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IMA Newsletter #409

November 2010

2010-2011 Program

See http://www.ima.umn.edu/2010-2011/ for a full description of the 2010-2011 program on Simulating Our Complex World: Modeling, Computation and Analysis.

2010-2011 IMA Participating Institutions Conferences

IMA Events

IMA Annual Program Year Workshop

Numerical Solutions of Partial Differential Equations: Novel Discretization Techniques

November 1-5, 2010

Organizers: Susanne C. Brenner (Louisiana State University), Claudio Canuto (Politecnico di Torino), Chi-Wang Shu (Brown University)

IMA Workshop

Finite Element Circus featuring a Scientific Celebration of Falk, Pasciak, and Wahlbin

November 5-6, 2010

Organizers: Jay Gopalakrishnan (University of Florida), Johnny Guzman (Brown University), Peter Monk (University of Delaware)

IMA Tutorial

Fast Solution Techniques

November 28-29, 2010

Organizers: Susanne C. Brenner (Louisiana State University), Robert Falgout (Lawrence Livermore National Laboratory), Ricardo H. Nochetto (University of Maryland)

IMA Annual Program Year Workshop

Numerical Solutions of Partial Differential Equations: Fast Solution Techniques

November 29 - December 3, 2010

Organizers: Susanne C. Brenner (Louisiana State University), Robert Falgout (Lawrence Livermore National Laboratory), Ricardo H. Nochetto (University of Maryland)
Schedule

Monday, November 1

All Day Morning Chair: Clint Dawson (University of Texas at Austin)
Afternoon Chair: Chi-Wang Shu (Brown University)
General announcements during the week: Susanne C. Brenner (Louisiana State University)
W11.1-5.10
8:15am-8:45am Registration and coffee Keller Hall 3-176 W11.1-5.10
8:45am-9:00am Welcome to the IMAFadil Santosa (University of Minnesota)Keller Hall 3-180 W11.1-5.10
9:00am-9:45am Hilbert complexes and the finite element exterior calculusDouglas N. Arnold (University of Minnesota)Keller Hall 3-180 W11.1-5.10
9:45am-10:30am Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin and finite volume schemes for conservation lawsChi-Wang Shu (Brown University)Keller Hall 3-180 W11.1-5.10
10:30am-11:00am Coffee break W11.1-5.10
11:00am-11:45am Multi-agent cooperative dynamical systems: Theory and numerical simulationsClaudio Canuto (Politecnico di Torino)Keller Hall 3-180 W11.1-5.10
11:45am-2:00pm Lunch W11.1-5.10
2:00pm-2:45pm Central DG methods for Hamilton-Jacobi equations and ideal MHD equationsFengyan Li (Rensselaer Polytechnic Institute)Keller Hall 3-180 W11.1-5.10
2:45pm-3:30pm The solution of time harmonic wave equations using complete families of elementary solutionsPeter Monk (University of Delaware)Keller Hall 3-180 W11.1-5.10
3:30pm-3:40pm Group photo W11.1-5.10
4:00pm-6:00pm Reception and Poster Session
Poster submissions welcome from all participants
Instructions
Lind Hall 409 W11.1-5.10
Space-time sparse wavelet FEM for parabolic equationsRoman Andreev (ETH Zürich)
Properties of the volume corrected characteristic mixed methodTodd Arbogast (University of Texas at Austin)
Pseudo-time continuation and time marching methods for Monge-Ampère type equationsGerard Michel Awanou (Northern Illinois University)
Spectral methods for systems of coupled equations and applications to Cahn-Hilliard equationsFeng Chen (Purdue University)
Reduced-order modelling for electromagneticsYanlai Chen (University of Massachusetts, Dartmouth)
Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equationsYingda Cheng (Brown University)
Hodge decomposition and Maxwell's equationsJintao Cui (University of Minnesota)
Solution of dual-mixed elasticity equations using Arnold-Falk-Winther element and discontinuous Petrov-Galerkin method. A comparisonLeszek Feliks Demkowicz (University of Texas at Austin)
Application of DPG method to wave propagationLeszek Feliks Demkowicz (University of Texas at Austin)
Application of DPG method to Stokes equationsLeszek Feliks Demkowicz (University of Texas at Austin)
Application of DPG method to hyperbolic problemsLeszek Feliks Demkowicz (University of Texas at Austin)
Adaptive solution of parametric eigenvalue problems for partial differential equations Joscha Gedicke (Humboldt-Universität)
Cochain interpolation for spectral element methodsMarc Iwan Gerritsma (Technische Universiteit te Delft)
Error estimates for generalized barycentric interpolationAndrew Kruse Gillette (University of Texas at Austin)
A C0 interior penalty method for a biharmonic problem with essential and natural boundary conditions of Cahn-Hilliard typeShiyuan Gu (Louisiana State University)
Multiscale methods for complex systemsAngela Kunoth (Universität Paderborn)
Local properties of finite element solutions for advection-dominated optimal control problems Dmitriy Leykekhman (University of Connecticut)
Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation lawsGuang Lin (Pacific Northwest National Laboratory)
Multiscale finite element method in perforated domainsAlexei Lozinski (Université de Toulouse III (Paul Sabatier))
Computational aspects of a two-scale finite element method for singularly perturbed problemsNiall Madden (National University of Ireland, Galway)
Wavenumber-explicit convergence analysis for the Helmholtz equation: hp-FEM and hp-BEMJens Markus Melenk (Technische Universität Wien)
Conformal conservation laws and geometric integration for Hamiltonian PDE with added dissipationBrian Edward Moore (University of Central Florida)
Finite element methods for the Monge-Ampere equationMichael Joseph Neilan (Louisiana State University)
HDG methods for multiphysics simulationNgoc-Cuong Nguyen (Massachusetts Institute of Technology)
HDG methods for CFD applications Ngoc-Cuong Nguyen (Massachusetts Institute of Technology)
The generalized fundamental theorem of calculus and its applications to boundary element methodsSylvain Nintcheu Fata (Oak Ridge National Laboratory)
High order integral deferred correction method based on Strang split semi-Lagrangian WENO method for Vlasov Poisson simulationsJingmei Qiu (Colorado School of Mines)
A reduced basis hybrid method for viscous flows in parametrized complex networksGianluigi Rozza (École Polytechnique Fédérale de Lausanne (EPFL))
Isogeometric Analysis for electromagnetic problemsGiancarlo Sangalli (Università di Pavia)
Numerical smoothness and error analysis for RKDG on the scalar nonlinear conservation lawsTong Sun (Bowling Green State University)
Numerical study of singular solutions of relativistic Euler equationsYi Sun (Statistical and Applied Mathematical Sciences Institute (SAMSI))
Efficient, accurate and rapidly-convergent algorithms for the solution of three dimensional acoustic and electromagnetic scattering problems in domains with geometric singularitiesCatalin Turc (Case Western Reserve University)
The flux reconstruction approach to high-order methods: Theory and applicationPeter Edward Vincent (Stanford University)
Shape optimization of chiral propellers in 3-D stokes flowShawn W. Walker (Louisiana State University)
High order well-balanced schemes for non-equilibrium flowsWei Wang (Florida International University)
GPU accelerated discontinuous Galerkin methodsTimothy C. Warburton (Rice University)
Numerical simulation of three-dimensional nonlinear water wavesLiwei Xu (Rensselaer Polytechnic Institute)
Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field Liwei Xu (Rensselaer Polytechnic Institute)
A multipoint flux mixed finite element method on general hexahedra: Multiscale mortar extension and applications to multiphase flow in porous mediaGuangri Xue (University of Texas at Austin)

Tuesday, November 2

All Day Morning Chair: Richard S. Falk (Rutgers University)
Afternoon Chair: Jan S. Hesthaven (Brown University)
W11.1-5.10
8:30am-9:00am CoffeeKeller Hall 3-176 W11.1-5.10
9:00am-9:45am Two grids approximation of non linear eigenvalue problemsYvon Jean Maday (Université de Paris VI (Pierre et Marie Curie))Keller Hall 3-180 W11.1-5.10
9:45am-10:30am Exterior calculus and the finite element approximation of Maxwell's eigenvalue problemDaniele Boffi (Università di Pavia)Keller Hall 3-180 W11.1-5.10
10:30am-11:00am Coffee break W11.1-5.10
11:00am-11:45am Trefftz-discontinuous Galerkin methods for time-harmonic wave problemsIlaria Perugia (Università di Pavia)Keller Hall 3-180 W11.1-5.10
11:45am-2:00pm Lunch W11.1-5.10
2:00pm-2:45pm New algorithms for high frequency wave propagationBjorn Engquist (University of Texas at Austin)Keller Hall 3-180 W11.1-5.10
2:45pm-3:30pm A new class of adaptive discontinuous Petrov-Galerkin (DPG) finite element (FE) methods with application to singularly perturbed problemsLeszek Feliks Demkowicz (University of Texas at Austin)Keller Hall 3-180 W11.1-5.10
3:30pm-4:00pm Coffee breakKeller Hall 3-176 W11.1-5.10
4:00pm-4:45pm Absolutely stable IPDG and LDG methods for high frequency wave equationsXiaobing Henry Feng (University of Tennessee)Keller Hall 3-180 W11.1-5.10

Wednesday, November 3

All Day Morning Chair: Ronald H.W. Hoppe (University of Houston)
Afternoon Chair: Ricardo H. Nochetto (University of Maryland)
W11.1-5.10
8:30am-9:00am CoffeeKeller Hall 3-176 W11.1-5.10
9:00am-9:45am Sparse tensor Galerkin discretizations for first order transport problemsChristoph Schwab (ETH Zürich)Keller Hall 3-180 W11.1-5.10
9:45am-10:30am New efficient spectral methods for high-dimensional PDEs and for Fokker-Planck equation of FENE dumbbell modelJie Shen (Purdue University)Keller Hall 3-180 W11.1-5.10
10:30am-11:00am Coffee break W11.1-5.10
11:00am-11:45am Stable enrichment and treatment of complex domains in the particle–partition of unity methodMarc Alexander Schweitzer (Rheinische Friedrich-Wilhelms-Universität Bonn)Keller Hall 3-180 W11.1-5.10
11:45am-2:00pm Lunch W11.1-5.10
2:00pm-2:45pm Adaptive tensor product wavelet methods for solving well-posed operator equationsRob Stevenson (Universiteit van Amsterdam)Keller Hall 3-180 W11.1-5.10
2:45pm-3:30pm Variational consistent discretization schemes and numerical algorithms for contact problemsBarbara Wohlmuth (Technical University of Munich )Keller Hall 3-180 W11.1-5.10
5:00pm-6:30pm Social "hour" Stub and Herbs
227 Oak St Minneapolis, MN 55414
Stub and Herbs
227 Oak St Minneapolis, MN 55414
(612) 379-0555
W11.1-5.10

Thursday, November 4

All Day Morning Chair: Lucia Gastaldi (Università di Brescia)
Afternoon Chair: Ragnar Winther (University of Oslo)
W11.1-5.10
8:30am-9:00am CoffeeKeller Hall 3-176 W11.1-5.10
9:00am-9:45am A different look at transport problemsWolfgang Dahmen (RWTH Aachen)Keller Hall 3-180 W11.1-5.10
9:45am-10:30am Space-time adaptive wavelet methods for control problems constrained by parabolic PDEsAngela Kunoth (Universität Paderborn)Keller Hall 3-180 W11.1-5.10
10:30am-11:00am Coffee break W11.1-5.10
11:00am-11:45am Isogeometric Analysis of Fluids, Structures and Fluid-Structure InteractionYuri Bazilevs (University of California, San Diego)Keller Hall 3-180 W11.1-5.10
11:45am-2:00pm Lunch W11.1-5.10
2:00pm-2:45pm Adaptive data analysis via nonlinear compressed sensingThomas Yizhao Hou (California Institute of Technology)Keller Hall 3-180 W11.1-5.10
2:45pm-3:30pm The adaptive Anisotropic PML method for time-harmonic acoustic and electromagnetic scattering problemsZhiming Chen (Chinese Academy of Sciences)Keller Hall 3-180 W11.1-5.10
3:30pm-4:00pm Coffee breakKeller Hall 3-176 W11.1-5.10
4:00pm-5:00pm Roundtable/Discussion
Panel Moderator: Chi-Wang Shu (Brown University)
Keller Hall 3-180 W11.1-5.10

Friday, November 5

All Day Chair: Claudio Canuto (Politecnico di Torino) W11.1-5.10
8:30am-9:00am CoffeeKeller Hall 3-176 W11.1-5.10
9:00am-9:45am Optimization-based computational modeling, or how to achieve better predictiveness with less complexityPavel B. Bochev (Sandia National Laboratories)Keller Hall 3-180 W11.1-5.10
9:45am-10:30am Finite element discretizations of the contact between two membranes Christine Bernardi (Université de Paris VI (Pierre et Marie Curie))Keller Hall 3-180 W11.1-5.10
10:30am-11:00am Coffee break W11.1-5.10
11:00am-11:45am HDG methods for second-order elliptic problemsBernardo Cockburn (University of Minnesota)Keller Hall 3-180 W11.1-5.10
11:45am-11:50am Closing remarksKeller Hall 3-180 W11.1-5.10
12:30pm-1:30pm Registration and coffeeKeller Hall 3-176 SW11.5-6.10
1:30pm-1:45pm Welcome to the IMAFadil Santosa (University of Minnesota)Keller Hall 3-180 SW11.5-6.10
1:45pm-3:15pm Contributed talks Keller Hall 3-180 SW11.5-6.10
3:15pm-3:45pm Group photo/coffee breakKeller Hall 3-176 SW11.5-6.10
3:45pm-4:45pm Contributed talksKeller Hall 3-180 SW11.5-6.10
4:45pm-5:30pm Plenary talk: On the lumped mass finite element method for parabolic problemsVidar Thomée (Chalmers University of Technology)Keller Hall 3-180 SW11.5-6.10
6:00pm-8:00pm Conference banquet at Campus ClubCampus Club
4th Floor Coffman Memorial Union
SW11.5-6.10

Saturday, November 6

8:30am-9:00am CoffeeKeller Hall 3-176 SW11.5-6.10
9:00am-9:45am Plenary talk: Analysis of a Cartesian PML approximation to an acoustic scattering problemJames H. Bramble (Texas A & M University)Keller Hall 3-180 SW11.5-6.10
9:45am-10:30am Contributed talksKeller Hall 3-180 SW11.5-6.10
10:30am-11:00am Coffee breakKeller Hall 3-176 SW11.5-6.10
11:00am-12:00pm Contributed talksKeller Hall 3-180 SW11.5-6.10
12:00pm-1:30pm Lunch Study areas, Keller Hall Atrium SW11.5-6.10
1:30pm-2:45pm Contributed talksKeller Hall 3-180 SW11.5-6.10
2:45pm-3:15pm Coffee breakKeller Hall 3-176 SW11.5-6.10
3:15pm-4:15pm Contributed talks EE/CS 3-180 SW11.5-6.10
4:15pm-5:00pm Plenary talk: Canonical families of finite elements Douglas N. Arnold (University of Minnesota)Keller Hall 3-180 SW11.5-6.10

Monday, November 8

10:45am-11:15am Coffee breakLind Hall 400

Tuesday, November 9

10:45am-11:15am Coffee breakLind Hall 400
11:15am-12:15pm Robust numerical solution of singularly perturbed problemsNiall Madden (National University of Ireland, Galway)Lind Hall 305 PS
7:00pm-8:00pm Arnold family lecture: Burst, cascades, and hot spots: A glimpse of some on-line social phenomena at global scalesJon Kleinberg (Cornell University)Willey Hall 175 PUB11.9.10

Wednesday, November 10

10:45am-11:15am Coffee breakLind Hall 400
11:15am-12:15pm The generalized finite element methodUday Banerjee (Syracuse University)Lind Hall 305

Thursday, November 11

10:45am-11:15am Coffee breakLind Hall 400
11:15am-12:15pm Special course: Finite element exterior calculusDouglas N. Arnold (University of Minnesota)Lind Hall 305

Friday, November 12

10:45am-11:15am Coffee breakLind Hall 400
11:15am-12:15pm The generalized finite element methodUday Banerjee (Syracuse University)Lind Hall 305
1:25pm-2:25pm A perspective on the use of mathematics in industryJeffrey Abell (General Motors)Vincent Hall 16 IPS

Monday, November 15

10:45am-11:15am Coffee breakLind Hall 400
2:00pm-3:30pm An introduction to the a posteriori error analysis of elliptic optimal control problemsRonald H.W. Hoppe (University of Houston)Lind Hall 305

Tuesday, November 16

10:45am-11:15am Coffee breakLind Hall 400
11:15am-12:15pm Discontinuous Galerkin approximation for the Vlasov-Poisson systemBlanca Ayuso de Dios (Centre de Recerca Matemàtica )Lind Hall 305 PS

Wednesday, November 17

10:45am-11:15am Coffee breakLind Hall 400
2:00pm-3:30pm An introduction to the a posteriori error analysis of elliptic optimal control problemsRonald H.W. Hoppe (University of Houston)Lind Hall 305

Thursday, November 18

10:45am-11:15am Coffee breakLind Hall 400
11:15am-12:15pm Special course: Finite element exterior calculusDouglas N. Arnold (University of Minnesota)Lind Hall 305

Friday, November 19

10:45am-11:15am Coffee breakLind Hall 400

Monday, November 22

10:45am-11:15am Coffee breakLind Hall 400

Tuesday, November 23

10:45am-11:15am Coffee breakLind Hall 400
11:15am-12:15pm TBASylvain Nintcheu Fata (Oak Ridge National Laboratory) PS

Wednesday, November 24

10:45am-11:15am Coffee breakLind Hall 400

Thursday, November 25

All Day Thanksgiving holiday. The IMA is closed.

Friday, November 26

All Day Floating holiday. The IMA is closed.

Sunday, November 28

All Day Morning Chair: Li-yeng Sung (Louisiana State University)
Afternoon Chair: Susanne C. Brenner(Louisiana State University)
T11.28-29.10
8:30am-9:00am Registration and coffeeKeller Hall 3-176 T11.28-29.10
9:00am-10:30am Geometric Multigrid MethodsSusanne C. Brenner (Louisiana State University)Keller Hall 3-180 T11.28-29.10
10:30am-11:00am BreakKeller Hall 3-176 T11.28-29.10
11:00am-12:30pm An algebraic multigrid tutorialRobert Falgout (Lawrence Livermore National Laboratory)Keller Hall 3-180 T11.28-29.10
12:30pm-2:00pm Lunch T11.28-29.10
2:00pm-3:30pm Domain decomposition methods for partial differential equationsDavid Keyes King Abdullah University of Science & Technology, Columbia UniversityKeller Hall 3-180 T11.28-29.10
3:30pm-4:00pm BreakKeller Hall 3-176 T11.28-29.10
4:00pm-5:30pm An introduction to domain decomposition algorithmsOlof B. Widlund (New York University)Keller Hall 3-180 T11.28-29.10

Monday, November 29

All Day Chair: Robert Falgout (Lawrence Livermore National Laboratory) T11.28-29.10
All Day Chair: Ulrich Rüde (Friedrich-Alexander-Universität Erlangen-Nürnberg)
General announcements during the week: Susanne C. Brenner (Louisiana State University)
W11.29-12.3.10
8:15am-8:45am CoffeeKeller Hall 3-176 T11.28-29.10
8:45am-10:15am Adaptive finite element methodsRicardo H. Nochetto (University of Maryland)Keller Hall 3-180 T11.28-29.10
10:15am-10:45am BreakKeller Hall 3-176 T11.28-29.10
10:45am-12:15pm A parallel computing tutorialUlrike Meier Yang (Lawrence Livermore National Laboratory)Keller Hall 3-180 T11.28-29.10
1:15pm-2:00pm Registration and coffeeKeller Hall 3-176 W11.29-12.3.10
2:00pm-2:15pm Welcome to the IMAFadil Santosa (University of Minnesota)Keller Hall 3-180 W11.29-12.3.10
2:15pm-3:00pm Some algorithmic aspects of hp-adaptive finite elementsRandolph E. Bank (University of California, San Diego)Keller Hall 3-180 W11.29-12.3.10
3:00pm-3:45pm Why it is so difficult to solve Helmholtz problems with iterative methodsMartin J. Gander (Universite de Geneve)Keller Hall 3-180 W11.29-12.3.10
3:45pm-4:00pm Group photo W11.29-12.3.10
4:00pm-5:30pm Reception and Poster Session
Poster submissions welcome from all participants
Instructions
Lind Hall 400 W11.29-12.3.10
Domain decomposition preconditioning for the hp-version of the discontinuous Galerkin methodPaola Francesca Antonietti (Politecnico di Milano)
Two-level additive Schwarz preconditioners for the local discontinuous Galerkin methodAndrew T. Barker (Louisiana State University)
Multigrid methods for two-dimensional Maxwell's equations on graded meshesJintao Cui (University of Minnesota)
Fast adaptive collocation by radial basis functionsTobin A. Driscoll (University of Delaware)
Adaptive solution of parametric eigenvalue problems for partial differential equationsJoscha Gedicke (Humboldt-Universität)
Adaptivity for the Hodge decomposition of Maxwell's equationsJoscha Gedicke (Humboldt-Universität)
An efficient rearrangement algorithm for shape optimization on eigenvalue problemsChiu-Yen Kao (Ohio State University)
Scalable electromagnetic simulations with the Auxiliary-space Maxwell Solver (AMS)Tzanio V Kolev (Lawrence Livermore National Laboratory)
Convergence and optimality of adaptive finite element methodsChristian Kreuzer (Universität Duisburg-Essen)
H-LU factorization of stabilized saddle point problemsSabine Le Borne (Tennessee Technological University)
A comparison of two domain decomposition methods for a linearized contact problemJungho Lee (Argonne National Laboratory)
3D boundary integral analysis by a precorrected fast Fourier transform algorithmSylvain Nintcheu Fata (Oak Ridge National Laboratory)
Two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty methodEun-Hee Park (Louisiana State University)
Energy minimization algebraic multigrid: Robustness and flexibility in multilevel softwareRay S. Tuminaro (Sandia National Laboratories)
Finite element analysis and a fast solver approach to a nonlocal dielectric continuum modelDexuan Xie (University of Wisconsin)
Hypre: A scalable linear solver library Ulrike Meier Yang (Lawrence Livermore National Laboratory)
A jumping multigrid method via finite element extrapolation Shangyou Zhang (University of Delaware)

Tuesday, November 30

All Day Morning Chair: Ronald H.W. Hoppe (University of Houston)
Afternoon Chair: Ulrike Meier Yang (Lawrence Livermore National Laboratory)
W11.29-12.3.10
8:30am-9:00am CoffeeKeller Hall 3-176 W11.29-12.3.10
9:00am-9:45am Fast Solvers for Higher Order ProblemsSusanne C. Brenner (Louisiana State University)Keller Hall 3-180 W11.29-12.3.10
9:45am-10:30am Nonsmooth Schur Newton Methods and ApplicationsRalf Kornhuber (Freie Universität Berlin)Keller Hall 3-180 W11.29-12.3.10
10:30am-11:00am Coffee breakKeller Hall 3-176 W11.29-12.3.10
11:00am-11:45am Nonlinear Multigrid RevisitedIrad Yavneh (Technion-Israel Institute of Technology)Keller Hall 3-180 W11.29-12.3.10
11:45am-2:00pm Lunch W11.29-12.3.10
2:00pm-2:45pm Multilevel iterative methods for PDEs based on one or no gridJinchao Xu (Pennsylvania State University)Keller Hall 3-180 W11.29-12.3.10
2:45pm-3:30pm Domain Decomposition Solvers for PDEs: Some Basics, Practical Tools, and New DevelopmentsClark R. Dohrmann (Sandia National Laboratories)Keller Hall 3-180 W11.29-12.3.10

Wednesday, December 1

All Day Morning Chair: Guido Kanschat (Texas A & M University)
Afternoon Chair: Ludmil Zikatanov (Pennsylvania State University)
W11.29-12.3.10
8:30am-9:00am CoffeeKeller Hall 3-176 W11.29-12.3.10
9:00am-9:45am Towards Exascale Computing: Multilevel Methods and Flow Solvers for Millions of CoresUlrich Rüde (Friedrich-Alexander-Universität Erlangen-Nürnberg)Keller Hall 3-180 W11.29-12.3.10
9:45am-10:30am Preconditioning Interior Penalty Discontinuous Galerkin MethodsBlanca Ayuso de Dios (Centre de Recerca Matemàtica )Keller Hall 3-180 W11.29-12.3.10
10:30am-11:00am Coffee breakKeller Hall 3-176 W11.29-12.3.10
11:00am-11:45am Preconditioners for interface problems in Eulerian formulations David Keyes King Abdullah University of Science & Technology, Columbia UniversityKeller Hall 3-180 W11.29-12.3.10
11:45am-2:00pm Lunch W11.29-12.3.10
2:00pm-2:45pm Compatible Relaxation in Algebraic MultigridRobert Falgout (Lawrence Livermore National Laboratory)Keller Hall 3-180 W11.29-12.3.10
2:45pm-3:30pm New Domain Decomposition Algorithms from OldOlof B. Widlund (New York University)Keller Hall 3-180 W11.29-12.3.10
5:00pm-6:30pm Social "hour" Stub and Herbs
227 Oak St Minneapolis, MN 55414
Stub and Herbs
227 Oak St Minneapolis, MN 55414
(612) 379-0555
W11.29-12.3.10

Thursday, December 2

All Day Morning Chair: Xiaobing Henry Feng (University of Tennessee)
Afternoon Chair: Uday Banerjee (Syracuse University)
W11.29-12.3.10
8:30am-9:00am CoffeeKeller Hall 3-176 W11.29-12.3.10
9:00am-9:45am Optimally Blended Spectral-Finite Element Scheme for Wave Propagation, and Non-Standard Reduced IntegrationMark Ainsworth (University of Strathclyde)Keller Hall 3-180 W11.29-12.3.10
9:45am-10:30am Local and global approximation of gradients with piecewise polynomialsAndreas Michael Veeser (Università di Milano)Keller Hall 3-180 W11.29-12.3.10
10:30am-11:00am Coffee breakKeller Hall 3-176 W11.29-12.3.10
11:00am-11:45am A Parallel, Adaptive, First-Order System Least-Squares (FOSLS) Algorithm for Incompressible, Resistive MagnetohydrodynamicsThomas A. Manteuffel (University of Colorado)Keller Hall 3-180 W11.29-12.3.10
11:45am-2:00pm Lunch W11.29-12.3.10
2:00pm-2:45pm Implicit Solution Approaches: Why We Care and How We Solve the SystemsCarol S. Woodward (Lawrence Livermore National Laboratory)Keller Hall 3-180 W11.29-12.3.10
2:45pm-3:30pm Coupled atmosphere - wildland fire numerical simulation by WRF-FireJan Mandel (University of Colorado)Keller Hall 3-180 W11.29-12.3.10
3:30pm-4:00pm Coffee breakKeller Hall 3-176 W11.29-12.3.10
4:00pm-5:00pm Roundtable/discussion
Moderator: Robert Falgout (Lawrence Livermore National Laboratory)
Keller Hall 3-180 W11.29-12.3.10

Friday, December 3

All Day Morning Chair: Axel Klawonn (Universität Duisburg-Essen)
Afternoon Chair: Susanne C. Brenner (Louisiana State University)
W11.29-12.3.10
8:30am-9:00am CoffeeKeller Hall 3-176 W11.29-12.3.10
9:00am-9:45am Developing fast and scalable implicit methods for shallow water equations on cubed-sphereXiao-Chuan Cai (University of Colorado)Keller Hall 3-180 W11.29-12.3.10
9:45am-10:30am Domain Decomposition for the Wilson Dirac OperatorAndreas Frommer (Bergische Universität-Gesamthochschule Wuppertal (BUGH))Keller Hall 3-180 W11.29-12.3.10
10:30am-11:00am Coffee breakKeller Hall 3-176 W11.29-12.3.10
11:00am-11:45am A posteriori error estimator competition for 2nd-order partial differential equations*Carsten Carstensen (Yonsei University)Keller Hall 3-180 W11.29-12.3.10
11:45am-2:00pm Lunch W11.29-12.3.10
2:00pm-2:45pm Projection based model reduction for shape optimization of the Stokes systemRonald H.W. Hoppe (University of Houston)Keller Hall 3-180 W11.29-12.3.10
2:45pm-3:30pm Convergence rates of AFEM with H -1 DataRicardo H. Nochetto (University of Maryland)Keller Hall 3-180 W11.29-12.3.10
Abstracts
Jeffrey Abell (General Motors) A perspective on the use of mathematics in industry
Abstract: A (possibly personal) perspective on how research that has a significant theoretical component should be constructed and organized. Questions regarding "what" and "why" will be explored with the group, along with a discussion of topics that are being investigated currently. Anyone interested in an interactive discussion is welcome to attend.
Mark Ainsworth (University of Strathclyde) Optimally Blended Spectral-Finite Element Scheme for Wave Propagation, and Non-Standard Reduced Integration
Abstract:
In an influential article, Marfurt suggested that the best scheme for computational wave propagation would involve an averaging of the consistent and lumped finite element approximations. Many authors have considered how this might be accomplished for first order approximation, but the case of higher orders remained unsolved.  We describe recent work on the dispersive and dissipative properties of a novel scheme for computational wave propagation obtained by averaging the consistent (finite element) mass matrix and lumped (spectral element) mass matrix. The objective is to obtain a hybrid scheme whose dispersive accuracy is superior to both of the schemes. We present the optimal value of the averaging constant for all orders of finite elements and proved that for this value the scheme is two orders more accurate compared with finite and spectral element schemes, and, in addition, the absolute accuracy is of this scheme is better than that of finite and spectral element methods.
 
Joint work with Hafiz Wajid, COMSATS Institute of Technology, Pakistan.
Roman Andreev (ETH Zürich) Space-time sparse wavelet FEM for parabolic equations
Abstract: For the model linear parabolic equation we propose a nonadaptive wavelet finite element space-time sparse discretization. The problem is reduced to a finite, overdetermined linear system of equations. We prove stability, i.e., that the finite section normal equations are well-conditioned if appropriate Riesz bases are employed, and that the Galerkin solution converges quasi-optimally in the natural solution space to the original equation. Numerical examples confirm the theory. This work is part of a PhD thesis under the supervision of Prof. Christoph Schwab, supported by Swiss National Science Foundation grant No. PDFMP2-127034/1.
Paola Francesca Antonietti (Politecnico di Milano) Domain decomposition preconditioning for the hp-version of the discontinuous Galerkin method
Abstract: We address the problem of efficiently solving the algebraic linear systems of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp discontinuous Galerkin finite element methods. We introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.

Joint work with Paul Houston, University of Nottingham, UK.
Todd Arbogast (University of Texas at Austin) Properties of the volume corrected characteristic mixed method
Abstract: Our goal is to accurately simulate transport of a miscible component in a bulk fluid over long times, i.e., locally conservatively and with little numerical diffusion. Characteristic methods have the potential to do this, since they have no CFL time-step constraints. The volume corrected characteristic mixed method was developed to conserve mass of both the component and the bulk fluid. We have proved that it has the important properties of monotonicity and stability, and therefore exhibits no overshoots nor undershoots. Moreover, the method converges optimally with or without physical diffusion. We show its performance through example simulations of pure curl flow and a nuclear waste repository.

Douglas N. Arnold (University of Minnesota) Hilbert complexes and the finite element exterior calculus
Abstract: The finite element exterior calculus, FEEC, has provided a viewpoint from which to understand and develop stable finite element methods for a variety of problems. It has enabled us to unify, clarify, and refine many of the classical mixed finite element methods, and has enabled the development of previously elusive stable mixed finite elements for elasticity. Just as an abstract Hilbert space framework helps clarify the theory of finite elements for model elliptic problems, abstract Hilbert complexes provides a natural framework for FEEC. In this talk we will survey the basic theory of Hilbert complexes and their discretization, discuss their applications to finite element methods. In particular, we will emphasize the role of two key properties, the subcomplex property and the bounded cochain projection property, in insuring stability of discretizations by transferring to the discrete level the structures that insure well-posedness of the PDE problem at the continuous level.
Douglas N. Arnold (University of Minnesota) Plenary talk: Canonical families of finite elements
Abstract: The most familiar family of finite elements is the Lagrange family, which provide the canonical finite element approximation of H1 on simplicial meshes in any dimension. In this talk we discuss families of simplicial and cubical finite elements—some previously known and some new—which are natural extensions of the Lagrange family in various ways. Even for some of the long known elements, a modern viewpoint based on the finite element exterior calculus provides new properties and insights.
Gerard Michel Awanou (Northern Illinois University) Pseudo-time continuation and time marching methods for Monge-Ampère type equations
Abstract: We discuss the performance of three numerical methods for the fully nonlinear Monge-Ampère equation. The first two are pseudo-time continuation methods while the third is a pure pseudo-time marching algorithm. The pseudo-time continuation methods are shown to converge for smooth data on a uniformly convex domain. We give numerical evidence that they perform well for the non-degenerate Monge-Ampère equation. The pseudo-time marching method applies in principle to any nonlinear equation. Numerical results with this approach for the degenerate Monge-Ampère equation are given as well as for the Pucci and Gauss-curvature equations.
Blanca Ayuso de Dios (Centre de Recerca Matemàtica ) Preconditioning Interior Penalty Discontinuous Galerkin Methods
Abstract:
We propose iterative methods for the solution of the linear systems resulting from
Interior Penalty (IP)  discontinuous Galerkin (DG)  approximations of elliptic problems.
The precoonditioners are derived from a natural decomposition of the DG finite element spaces. We present the convergence analysis of the solvers for both symmetric and non-symmetric IP schemes. Extension to problems with jumps in the coefficients and linear elasticity will also be discussed. We describe in detail the preconditioning techniques for low order (piece-wise linear ) IP methods and we indicate how to proceed in the case of high order (odd degree) approximations. The talk is based on joint works with Ludmil T. Zikatanov from  Penn State University (USA).
Blanca Ayuso de Dios (Centre de Recerca Matemàtica ) Discontinuous Galerkin approximation for the Vlasov-Poisson system
Abstract: One of the simplest model problems in the kinetic theory of plasma--physics is the Vlasov-Poisson (VP) system with periodic boundary conditions. Such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and self-consistent electric field. In this talk, we construct a new family of semi-discrete numerical schemes for the approximation of the Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin (DG) approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We present the error analysis in the case of smooth solutions. The issue of energy conservation is also analyzed for some of the methods. If time allows, I will also comment on the issue of approximating non-smooth solutions of the VP system. The talk is based on joint works with J.A. Carrillo (Universidad Autonoma de Barcelona) and C-W. Shu (Brown University).
Uday Banerjee (Syracuse University) The generalized finite element method
Abstract: The Generalized Finite Element Method (GFEM) is class of fexible Galerkin methods to approximate the solutions of partial differential equations. It allows to incorporate the "features" of the unknown solution into the trial space. This is done locally based on the available information about the unknown solution, which is often incomplete and "fuzzy." GFEM (in its various forms) has been used extensively in the engineering community to address problems involving cracks, interfaces, and certain micro-structures. In these talks, we will present a general survey of various aspects of the GFEM in the context of elliptic problems. We will discuss examples of trial spaces tailored to individual problems, together with their approximation properties. We will also address the condition number of the associated stiffness matrix, which can be a major issue in the GFEM. Some open problems in this area will also be highlighted in these lectures.
Randolph E. Bank (University of California, San Diego) Some algorithmic aspects of hp-adaptive finite elements
Abstract: We will discuss our on-going investigation of hp-adaptive finite elements. We will focus on a posteriori error estimates based on superconvergent derivative recovery. Besides providing both global error estimates and local error indicators, this family of error estimates also provides information that forms the basis of our hp-adaptive refinement and coarsening strategies. In particular, these a posteriori error estimates address in a cost efficient and natural way the critical issue of deciding between h or p refinement/coarsening. Some numerical examples will be provided.
Andrew T. Barker (Louisiana State University) Two-level additive Schwarz preconditioners for the local discontinuous Galerkin method
Abstract: We propose and analyze two-level overlapping additive Schwarz preconditioners for the local discontinuous Galerkin discretization. We prove a condition number estimate and show numerically that the method is scalable in terms of linear iterations. We also present numerical evidence that a parallel implementation of the method shows good scalability and speedup.
Yuri Bazilevs (University of California, San Diego) Isogeometric Analysis of Fluids, Structures and Fluid-Structure Interaction
Abstract:

Isogeometric Analysis [1] is a recently developed novel discratization technique that is based on the basis functions of computer-aided design and computer graphics. Although the main motivation behind the development of Isogeometric Analysis was to establish a tighter link between geometry modeling and computational analysis procedures, the new technology demonstrated better per-degree-of-freedom performance than standard finite elements on a broad range of problems in computational mechanics. This better "efficiency" of isogeometric analysis was attributed to more accurate analysis geometry definition and higher-order smoothness of the underlying basis functions. In this presentation, I will give an overview of the early developments in isogeometric analysis of fluids and structures. I will aslo give a summary of approximation results for the function spaces employed in isogeometric analysis. In the main body of the presentation I will show our recent work on isogeometric shell structures, turbulence modeling and fluid-structure interaction (FSI). I will conclude by presenting our recent isogeometric FSI simulations of a wind turbine rotor operating under realistic wind conditions and at full spatial scale in 3D [2,3].

 References

[1] J.A. Cottrell, T.J.R. Hughes, and Y. Bazilevs, “Isogeometric Analysis. Toward Integration of CAD and FEA”, Wiley 2009.

[2] Y. Bazilevs, M.-C. Hsu, I. Akkerman, S. Wright, K. Takizawa, B. Henicke, T. Spielman, and T.E. Tezduyar, “3D Simulation of Wind Turbine Rotors at Full Scale. Part I: Geometry Modeling and Aerodynamics”, International Journal of Numerical Methods in Fluids, (2010). Published online.

[3] Y. Bazilevs, M.-C. Hsu, J. Kiendl, R. Wuechner and K.-U. Bletzinger, “3D Simulation of Wind Turbine Rotors at Full Scale. Part II: Fluid-Structure Interaction”, International Journal of Numerical Methods in Fluids, (2010). Accepted for publcation.

 

Christine Bernardi (Université de Paris VI (Pierre et Marie Curie)) Finite element discretizations of the contact between two membranes
Abstract: The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. We first propose a discretization of this system, where the displacements are approximated by standard finite elements and the action by a local postprocessing which admits an equivalent mixed reformulation. We perform the a posteriori analysis of this discretization and prove optimal error estimates. Numerical experiments confirm the efficiency of the error indicators.
Pavel B. Bochev (Sandia National Laboratories) Optimization-based computational modeling, or how to achieve better predictiveness with less complexity
Abstract: Discretization converts infinite dimensional mathematical models into finite dimensional algebraic equations that can be solved on a computer. This process is accompanied by unavoidable information losses which can degrade the predictiveness of the discrete equations. Compatible and regularized discretizations control these losses directly by using suitable field representations and/or by modifications of the variational forms. Such methods excel in controlling "structural" information losses responsible for the stability and well-posedness of the discrete equations. However, direct approaches become increasingly complex and restrictive for multi-physics problems comprising of fundamentally different mathematical models, and when used to control losses of "qualitative" properties such as maximum principles, positivity, monotonicity and local bounds preservation.

In this talk we show how optimization ideas can be used to control externally, and with greater flexibility, information losses which are difficult (or impractical) to manage directly in the discretization process. This allows us to improve predictiveness of computational models, increase robustness and accuracy of solvers, and enable efficient reuse of code. Two examples will be presented: an optimization-based framework for multi-physics coupling, and an optimization-based algorithm for constrained interpolation (remap). In the first case, our approach allows to synthesize a robust and efficient solver for a coupled multiphysics problem from simpler solvers for its constituent components. To illustrate the scope of the approach we derive such a solver for nearly hyperbolic PDEs from standard, off-the-shelf algebraic multigrid solvers, which by themselves cannot solve the original equations. The second example demonstrates how optimization ideas enable design of high-order conservative, monotone, bounds preserving remap and transport schemes which are linearity preserving on arbitrary unstructured grids, including grids with polyhedral and polygonal cells.

This is a joint work with D. Ridzal , G. Scovazzi (SNL) and M. Shashkov (LANL).


Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin company, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
Daniele Boffi (Università di Pavia) Exterior calculus and the finite element approximation of Maxwell's eigenvalue problem
Abstract: Maxwell's eigenvalue problem can be seen as a particular case of the Hodge-Laplace eigenvalue problem in the framework of exterior calculus. In this context we present two mixed formulations that are equivalent to the problem under consideration and their numerical approximation. It turns out that the natural conditions for the good approximation of the eigensolutions of the mixed formulations are equivalent to a well-known discrete compactness property that has been firstly used by Kikuchi for the analysis of edge finite elements.

The result can be applied to the convergence analysis of the p-version of edge finite elements for the approximation of Maxwell's eigenvalue problem.
James H. Bramble (Texas A & M University) Plenary talk: Analysis of a Cartesian PML approximation to an acoustic scattering problem
Abstract: We consider a Cartesian PML approximation to solutions of acoustic scattering problems on an unbounded domain in ℝ2 and ℝ3. The perfectly matched layer (PML) technique modifies the equations outside of a bounded domain containing the region of interest. This is done in such a way that the new problem (still on an unbounded domain) has a solution which agrees with the solution of the original problem. The new problem has a solution which decays much faster, thus suggesting replacing it by a problem on a bounded domain. The perfectly matched layer (PML) technique, in a curvilinear coordinate system and in Cartesian coordinates, has been studied for acoustic scattering applications both in theory and computation. Using a different approach we extend the results of Kim and Pasciak concerning the PML technique in Cartesian coordinates. The exponential convergence of approximate solutions as a function of domain size and/or the PML "strength" parameter, σ0 is also shown. We note that once the stability and convergence of the (continuous) truncated problem has been established, the analysis of the resulting finite element approximations is then classical. Finally, the results of numerical computations illustrating the theory, in terms of efficiency and parameter dependence of the Cartesian PML approach will be given.
Susanne C. Brenner (Louisiana State University) Geometric Multigrid Methods
Abstract:
Geometric multigrid methods solve an elliptic boundary value
problem on a sequence of grids generated by a refinement
procedure.  They have optimal complexity in the sense that
the computational cost is proportional to the number of
unknowns.  In this tutorial we will introduce various
multigrid algorithms (V-cycle, W-cycle, F-cycle, etc.)
and discuss their convergence analysis.
Susanne C. Brenner (Louisiana State University) Fast Solvers for Higher Order Problems
Abstract:
There are two main difficulties in solving higher order elliptic
boundary value problems: the discretization schemes are more
complicated and the discrete problems are very ill-conditioned. 
In this talk we will discuss an approach that can solve higher
order problems with an efficiency similar to that for second order
problems. It is based on discontinuous Galerkin methods, embedded
multigrid algorithms and domain decomposition techniques.
 
Xiao-Chuan Cai (University of Colorado) Developing fast and scalable implicit methods for shallow water equations on cubed-sphere
Abstract: We are interested in solving coupled systems of partial differential equations on computers with a large number of processors. With some combinations of domain decomposition and multigrid methods, one can easily design algorithms that are highly scalable in terms of the number of linear and nonlinear iterations. However, if the goal is to minimize the total compute time and keep the near ideal scalability at the same time, then the task is more difficult. We discuss some recent experience in solving the shallow water equations on the sphere for the modeling of the global climate.

This is a joint work with C. Yang.
Claudio Canuto (Politecnico di Torino) Multi-agent cooperative dynamical systems: Theory and numerical simulations
Abstract: We are witnessing an increasing interest for cooperative dynamical systems proposed in the recent literature as possible models for opinion dynamics in social and economic networks. Mathematically, they consist of a large number, N, of 'agents' evolving according to quite simple dynamical systems coupled in according to some 'locality' constraint. Each agent i maintains a time function xi(t) representing the 'opinion,' the 'belief' it has on something. As time elapses, agent i interacts with neighbor agents and modifies its opinion by averaging it with the one of its neighbors. A critical issue is the way 'locality' is modelled and interaction takes place. In Krause's model each agent can see the opinion of all the others but averages with only those which are within a threshold R from its current opinion.

The main interest for these models is for N quite large. Mathematically, this means that one takes the limit for N → + ∞. We adopt an Eulerian approach, moving focus from opinions of various agents to distributions of opinions. This leads to a sort of master equation which is a PDE in the space of probabily measures; it can be analyzed by the techniques of Transportation Theory, which extends in a very powerful way the Theory of Conservation Laws. Our Eulerian approach gives rise to a natural numerical algorithm based on the `push forward' of measures, which allows one to perform numerical simulations with complexity independent on the number of agents, and in a genuinely multi-dimensional manner.

We prove the existence of a limit measure as t → ∞, which for the exact dynamics is purely atomic with atoms at least at distance R apart, whereas for the numerical dynamics it is 'almost purely atomic' (in a precise sense). Several representative examples will be discussed.

This is a joint work with Fabio Fagnani and Paolo Tilli.
Carsten Carstensen (Yonsei University) A posteriori error estimator competition for 2nd-order partial differential equations*
Abstract: Five classes of up to 13 a posteriori error estimators compete in three second-order model cases, namely the conforming and non-conforming first-order approximation of the Poisson-Problem plus some conforming obstacle problem. Since it is the natural first step, the error is estimated in the energy norm exclusively — hence the competition has limited relevance. The competition allows merely guaranteed error control and excludes the question of the best error guess. Even nonsmooth problems can be included. For a variational inequality, Braess considers Lagrange multipliers and some resulting auxiliary equation to view the a posteriori error control of the error in the obstacle problem as computable terms plus errors and residuals in the auxiliary equation. Hence all the former a posteriori error estimators apply to this nonlinear benchmark example as well and lead to surprisingly accurate guaranteed upper error bounds. This approach allows an extension to more general boundary conditions and a discussion of efficiency for the affine benchmark examples. The Luce-Wohlmuth and the least-square error estimators win the competition in several computational benchmark problems. Novel equilibration of nonconsistency residuals and novel conforming averaging error estimators win the competition for Crouzeix-Raviart nonconforming finite element methods. Our numerical results provide sufficient evidence that guaranteed error control in the energy norm is indeed possible with efficiency indices between one and two. Furthermore, accurate error control is slightly more expensive but pays off in all applications under consideration while adaptive mesh-refinement is sufficiently pleasant as accurate when based on explicit residual-based error estimates. Details of our theoretical and empirical ongoing investigations will be found in the papers quoted below.

References:
  1. S. Bartels, C. Carstensen, R. Klose, An experimental survey of a posteriori Courant finite element error control for the Poisson equation, Adv. Comput. Math., 15 (2001), pp. 79-106.
  2. C. Carstensen, C. Merdon, Estimator competition for Poisson problems, J. Comp. Math., 28 (2010), pp. 309-330.
  3. C. Carstensen, C. Merdon, Computational survey on a posteriori error estimators for nonconforming finite element methods, Part I: Poisson problems (in preparation)
  4. C. Carstensen, C. Merdon, A posteriori error estimator competition for conforming obstacle problems, Part I: Theoretical findings (in preparation), Part II: Numerical results (in preparation)

* This work was supported by DFG Research Center MATHEON and by the World Class University (WCU) program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology R31-2008-000-10049-0.
Feng Chen (Purdue University) Spectral methods for systems of coupled equations and applications to Cahn-Hilliard equations
Abstract: I will present how our new developed spectral method solvers can be applied to highly nonlinear and high-order evolution equations such as strongly anisotropic Cahn-Hilliard equations from materials science. In addition, we consider how to design schemes that are energy stable and easy to solve (avoid solving nonlinear equations implicitly). We use the Legendre-Galerkin method to simulate the anisotropic Cahn-Hilliard equation with the Willmore regularization. Excellent agreement between numerical simulations and theoretical results are observed.
Yanlai Chen (University of Massachusetts, Dartmouth) Reduced-order modelling for electromagnetics
Abstract: The reduced basis method (RBM) is indispensable in scenarios where a large number of solutions to a parametrized partial differential equation are desired. These include simulation-based design, parameter optimization, optimal control, multi-model/scale simulation etc. Thanks to the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimentional spaces, RBM can improve efficiency reliably by several orders of magnitudes. This poster presents RBM for various electromagnetic problems including radar cross section computation of an object whose scattered field is highly sensitive to the geometry. We also propose a new reduced basis element method (RBEM) that simulate electromagnetic wave propagation in a pipe of varying shape. This is joint work with Jan Hesthaven and Yvon Maday.
Zhiming Chen (Chinese Academy of Sciences) The adaptive Anisotropic PML method for time-harmonic acoustic and electromagnetic scattering problems
Abstract: We report our recent efforts in developing the adaptive perfectly matched layer (PML) method solving the time-harmonic electromagnetic and acoustic scattering problems. The PML parameters such as the thickness of the layer and the absorbing medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the adaptive PML method provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the choice of the thickness of the PML layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.
Yingda Cheng (Brown University) Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations
Abstract: We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (BTE) under the action of quadratically confined electrostatic potentials. The solutions of the BTEs are positive probability distribution functions. It is very challenging to have a mass-conservative, high-order accurate scheme that preserves positivity of the numerical solutions in high dimensions. Our work extends the maximum-principle-satisfying scheme for scalar conservation laws to include the linear Boltzmann collision term. The DG schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy. A discussion of the standard semi-discrete DG schemes for the BTE are included as a foundation for the stability and error estimates for this new scheme. Numerical results of the relaxation models are provided to validate the method.

Bernardo Cockburn (University of Minnesota) HDG methods for second-order elliptic problems
Abstract:

In this talk, we discuss a new class of discontinuous Galerkin methods called "hybridizable". Their distinctive feature is that the only globally-coupled degrees of freedom are those of the numerical trace of the scalar variable. This renders them efficiently implementable. Moreover, they are more precise than all other discontinuous Galerkin methods as thet share with mixed methods their superconvergence properties in the scalar variable and their optimal order of convergence for the vector variable. We are going to show how to devise these methods and comment on their implementation and convergence properties.

Jintao Cui (University of Minnesota) Hodge decomposition and Maxwell's equations
Abstract: In this work we investigate the numerical solution for two-dimensional Maxwell's equations on graded meshes. The approach is based on the Hodge decomposition for divergence-free vector fields. An approximate solution for Maxwell's equations is obtained by solving standard second order elliptic boundary value problems. We illustrate this new approach by a P1 finite element method.
Jintao Cui (University of Minnesota) Multigrid methods for two-dimensional Maxwell's equations on graded meshes
Abstract: In this work we investigate the numerical solution for two-dimensional Maxwell's equations on graded meshes. The approach is based on the Hodge decomposition. The solution u of Maxwell's equations is approximated by solving standard second order elliptic problems. Quasi-optimal error estimates for both u and curl of u in the L2 norm are obtained on graded meshes. We prove the uniform convergence of the W-cycle and full multigrid algorithms for the resulting discrete problem.
Wolfgang Dahmen (RWTH Aachen) A different look at transport problems
Abstract: Joint work with Chunyan Huang, Christoph Schwab and Gerrit Welper.

The success of adaptive (wavelet) methods for operator equations relies on well-posedness of suitable variational formulations and on the availability of Riesz bases (or frames) for the corresponding energy space provided that the corresponding representation of the operator is in a certain sense quasi sparse. When dealing with transport dominated problems such favorable conditions are no longer met for the commonly used variational principles. Moreover, solutions typically exhibit strongly anisotropic features such as layers or shocks. Focussing for simplicity on the simplest model of linear transport we present alternative variational formulations that are, in particular, stable in L2 so that corresponding discrete solutions are best approximants in L2. Moreover, this provides a theoretical platform for ultimately employing directional representation systems like shearlets, which are known to form L2-frames and offer much more economical sparse representations of anisotropic structures than classical wavelet systems. This is a central objective in an ongoing collaboration with G. Kutyniok's group within the Priority Research Programme (SPP) No. 1324 of the German Research Foundation. In principle, the approach can be understood as a Petrov-Galerkin formulation in the infinite dimensional setting. We address several theoretical and (uncommon) numerical tasks arising in this context and indicate first steps towards rigorously founded adaptive solution concepts. These results are illustrated by preliminary numerical experiments first in a finite element setting.
Leszek Feliks Demkowicz (University of Texas at Austin) A new class of adaptive discontinuous Petrov-Galerkin (DPG) finite element (FE) methods with application to singularly perturbed problems
Abstract: Joint work with Jay Golapalakrishnan, U. Florida.

Adaptive finite elements vary element size h or/and polynomial order p to deliver approximation properties superior to standard discretization methods. The best approximation error may converge even exponentially fast to zero as a function of problem size (CPU time, memory). The adaptive methods are thus a natural candidate for singularly perturbed problems like convection-dominated diffusion, compressible gas dynamics, nearly incompressible materials, elastic deformation of structures with thin-walled components, etc. Depending upon the problem, diffusion constant, Poisson ratio or beam (plate, shell) thickness, define the small parameter.

This is the good news. The bad news is that only a small number of variational formulations is stable for adaptive meshes By the stability we mean a situation where the discretization error can be bounded by the best approximation error times a constant that is independent of the mesh. To this class belong classical elliptic problems (linear and non-linear), and a large class of wave propagation problems whose discretization is based on hp spaces reproducing the classical exact grad-curl-div sequence. Examples include acoustics, Maxwell equations, elastodynamics, poroelasticity and various coupled and multiphysics problems.

For singularly perturbed problems, the method should also be robust, i.e. the stability constant should be independent of the perturbation parameter. This is also the dream for wave propagation problems in the frequency domain where the (inverse of) frequency can be identified as the perturbation parameter. In this context, robustness implies a method whose stability properties do not deteriorate with the frequency (method free of pollution (phase) error).

We will present a new paradigm for constructing discretization schemes for virtually arbitrary systems of linear PDE's that remain stable for arbitrary hp meshes, extending thus dramatically the applicability of hp approximations. The DPG methods build on two fundamental ideas:

- a Petrov-Galerkin method with optimal test functions for which continuous stability automatically implies discrete stability,

- a discontinuous Petrov-Galerkin formulation based on the so-called ultra-weak variational hybrid formulation.

We will use linear acoustics and convection-dominated diffusion as model problems to present the main concepts and then review a number of other applications for which we have collected some numerical experience including: 1D and 2D convection-dominated diffusion (boundary layers) 1D Burgers and compressible Navier-Stokes equations (shocks) Timoshenko beam and axisymmetric shells (locking, boundary layers) 2D linear elasticity (mixed formulation, singularities) 1D and 2D wave propagation (pollution error control) 2D convection and 2D compressible Euler equations (contact discontinuities and shocks)

The presented methodology incorporates the following features:

The problem of interest is formulated as a system of first order PDE's in the distributional (weak) form, i.e. all derivatives are moved to test functions. We use the DG setting, i.e. the integration by parts is done over individual elements.

As a consequence, the unknowns include not only field variables within elements but also fluxes on interelement boundaries. We do not use the concept of a numerical flux but, instead, treat the fluxes as independent, additional unknowns (a hybrid method).

For each trial function corresponding to either field or flux variable, we determine a corresponding optimal test function by solving an auxiliary local problem on one element. The use of optimal test functions guarantees attaining the supremum in the famous inf-sup condition from Babuska-Brezzi theory. The resulting stiffness matrix is always hermitian and positive-definite. In fact, the method can be interpreted as a least-squares applied to a preconditioned version of the problem.

By selecting right norms for test functions, we can obtain stability properties uniform not only with respect to discretization parameters but also with respect to the perturbation parameter (diffusion constant, Reynolds number, beam or shell thickness, wave number) In other words, the resulting discretization is robust.

For a detailed presentation on the subject, see [1-8].

[1] L. Demkowicz and J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods. Part I: The Transport Equation. Comput. Methods Appl. Mech. Engrg., in print. see also ICES Report 2009-12.

[2] L. Demkowicz and J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions. Numer. Mth. Partt. D.E., accepted, ICES Report 2009-16.

[3] L. Demkowicz, J. Gopalakrishnan and A. Niemi. A Class of Discontinuous Petrov-Galerkin Methods. Part III: Adaptivity. ICES Report 2010-1, submitted to ApNumMath.

[4] A. Niemi, J. Bramwell and L. Demkowicz, "Discontinuous Petrov-Galerkin Method with Optimal Test Functions for Thin-Body Problems in Solid Mechanics," ICES Report 2010-13, submitted to CMAME.

[5] J. Zitelli, I. Muga, L, Demkowicz, J. Gopalakrishnan, D. Pardo and V. Calo, "A class of discontinuous Petrov-Galerkin methods. IV: Wave propagation problems, ICES Report 2010-17, submitted to J.Comp. Phys.

[6] J. Bramwell, L. Demkowicz and W. Qiu, "Solution of Dual-Mixed Elasticity Equations Using AFW Element and DPG. A Comparison" ICES Report 2010-23.

[7] J. Chan, L. Demkowicz, R. Moser and N Roberts, "A class of Discontinuous Petrov-Galerkin methods. Part V: Solution of 1D Burgers and Navier--Stokes Equations" ICES Report 2010-25.

[8] L Demkowicz and J. Gopalakrishnan, "A Class of Discontinuous Petrov-Galerkin Methods. Part VI: Convergence Analysis for the Poisson Problem," ICES Report, in preparation.
Leszek Feliks Demkowicz (University of Texas at Austin) Solution of dual-mixed elasticity equations using Arnold-Falk-Winther element and discontinuous Petrov-Galerkin method. A comparison
Abstract: Joint work with J. Bramwell and W. Qiu.

The presentation is devoted to a numerical comparison and illustration of the two methods using a couple of 2D numerical examples. We compare stability properties of both methods and their efficiency.
Leszek Feliks Demkowicz (University of Texas at Austin) Application of DPG method to wave propagation
Abstract: Joint work with J. Zitelli, I. Muga, J. Gopalakrishnan, D. Pardo, and V. M. Calo.

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase error. 1D proof and both 1D and 2D numerical experiments are presented.
Leszek Feliks Demkowicz (University of Texas at Austin) Application of DPG method to Stokes equations
Abstract: Joint work with N.V. Roberts, D. Ridzal, P. Bochev, K.J. Peterson, and Ch. M. Siefert.

The DPG method of Demkowicz and Gopalakrishnan guarantees the optimality of the solution in what they call the energy norm. An important choice that must be made in the application of the method is the definition of the inner product on the test space. In te presentation we apply the DPG method to the Stokes problem in two dimensions, analyzing it to determine appropriate inner products, and perform a series of numerical experiments.
Leszek Feliks Demkowicz (University of Texas at Austin) Application of DPG method to hyperbolic problems
Abstract: Joint work with J. Chan.

We present an application of the DPG method to convection, linear systems of hyperbolic equations and the compressible Euler equations.
Clark R. Dohrmann (Sandia National Laboratories) Domain Decomposition Solvers for PDEs: Some Basics, Practical Tools, and New Developments
Abstract:

The first part of this talk provides a basic introduction to the building blocks of domain decomposition solvers. Specific details are given for both the classical overlapping Schwarz (OS) algorithm and a recent iterative substructuring (IS) approach called balancing domain decomposition by constraints (BDDC). A more recent hybrid OS-IS approach is also described.

The success of domain decomposition solvers depends critically on the coarse space. Similarities and differences between the coarse spaces for OS and BDDC approaches are discussed, along with how they can be obtained from discrete harmonic extensions. Connections are also made between coarse spaces and multiscale modeling approaches from computational mechanics. As a specific example, details are provided on constructing coarse spaces for incompressible fluid problems.

The next part of the talk deals with a variety of implementation details for domain decomposition solvers.  These include mesh partitioning options, local and global solver options, reducing the coarse space dimension, dealing with constraint equations, residual weighting to accelerate the convergence of OS methods, and recycling of Krylov spaces to efficiently solve problems with multiple right hand sides. Some potential bottlenecks and remedies for domain decomposition solvers are also discussed.

The final part of the talk concerns some recent theoretical advances, new algorithms, and open questions in the analysis of domain decomposition solvers. The focus will be primarily on the work of the speaker and his colleagues on elasticity, fluid mechanics, problems in H(curl), and the analysis of subdomains with irregular boundaries. The speaker gratefully acknowledges contributions of Jan Mandel and Olof Widlund to many topics discussed in this talk.

Tobin A. Driscoll (University of Delaware) Fast adaptive collocation by radial basis functions
Abstract: Radial basis functions provide flexible, meshfree approximations to functions and solutions of differential equations. Naive algorithms suffer from dense linear algebra and severe ill conditioning. Simple multiscale adaptive techniques for the nodes and shape parameters have previously proven very effective in controlling ill conditioning for small node sets. We present a new fast summation method suitable for adaptively generated basis functions with varying shape parameters. When coupled with an easily parallelized restricted additive Schwarz preconditioner, the method can find RBF coefficients in near O(N log N) time for N nodes.
Bjorn Engquist (University of Texas at Austin) New algorithms for high frequency wave propagation
Abstract: We will give a brief overview of multiscale modeling for wave equation problems and then focus on two techniques. One is an energy conserving DG method for time domain and the other is a new a new sweeping preconditioner for frequency domain simulation. The latter is resulting in a computational procedure that essentially scales linearly even in the high frequency regime.
Robert Falgout (Lawrence Livermore National Laboratory) An algebraic multigrid tutorial
Abstract: Multigrid methods are so-called optimal methods because they can solve a system of N unknowns with O(N) work. This optimality property is crucial for scaling up to huge high-resolution simulations on parallel computers. To achieve this, the multigrid components must be designed with the underlying system in mind, traditionally, the problem geometry. Algebraic multigrid, however, is a method for solving linear systems using multigrid principles, but requiring no explicit geometric information. Instead, AMG determines the essential multigrid ingredients based solely on the matrix entries.

Since the method's introduction in the mid-eighties, researchers have developed numerous AMG algorithms with different robustness and efficiency properties that target a variety of problem classes. In this tutorial, we will introduce the AMG method, beginning with a description of the classical algorithm of Achi Brandt, Steve McCormick, John Ruge, and Klaus Stüben, and then move on to more recent advances and theoretical developments.
Robert Falgout (Lawrence Livermore National Laboratory) Compatible Relaxation in Algebraic Multigrid
Abstract:
Algebraic multigrid (AMG) is an important method for solving the large sparse linear systems that arise in many PDE-based scientific simulation codes.  A major component of algebraic multigrid methods is the selection of coarse grids and the construction of interpolation.  The idea of compatible relaxation (CR) was introduced by Brandt as a cheap method for measuring the quality of a coarse grid.  In this talk, we will review the theory behind CR, describe our CR-based coarsening algorithm, and discuss aspects of the method that require additional development such as coarsening for systems of PDEs.  We will also discuss CR's ability to predict the convergence behavior of the AMG method and ideas for improving the accuracy of its predictions.  Finally, we will talk about issues of parallelizing these methods to run on massively parallel computers.
Xiaobing Henry Feng (University of Tennessee) Absolutely stable IPDG and LDG methods for high frequency wave equations
Abstract: In this talk I shall discuss some recent progresses in developing interior penalty discontinuous Galerkin (IPDG) methods and local discontinuous Galerkin (LDG) methods for high frequency scalar wave equation. The focus of the talk is to present some non-standard (h- and hp-) IPDG and LDG methods which are proved to be absolutely stable (with respect to the wave number and the mesh size) and optimally convergent (with respect to the mesh size). The proposed IPDG and LDG methods are shown to be superior over standard finite element and finite difference methods, which are known only to be stable under some stringent mesh constraints. In particular, it is observed that these non-standard IPDG and LDG methods are capable to correctly track the phases of the highly oscillatory waves even when the mesh violates the "rule-of-thumb" condition. Numerical experiments will be presented to show the efficiency of the non-standard IPDG and LDG methods. If time permits, latest generalizations of these DG methods to the high frequency Maxwell equations will also be discussed. This is a joint work with Haijun Wu of Nanjing University (China) and Yulong Xing of the University of Tennessee and Oak Ridge National Laboratory.
Andreas Frommer (Bergische Universität-Gesamthochschule Wuppertal (BUGH)) Domain Decomposition for the Wilson Dirac Operator
Abstract:
In lattice QCD, a standard discretization of the Dirac operator is given by the Wilson-Dirac operator, representing a nearest neighbor coupling on a 4d torus with 3x4 variables per grid point. The operator is non-symmetric but (usually) positive definite. Its small eigenmodes are non-smooth due to the stochastic nature of the coupling coefficients. Solving systems with the Wilson-Dirac operator on state-of-the-art lattices, typically in the range of 32-64 grid points in each of the four dimensions, is one of the prominent supercomputer applications today.
In this talk we will describe our experience with the domain decomposition principle as one approach to solve the Wilson-Dirac equation in parallel. We will report results on scaling with respect to the size of the overlap, on deflation techniques that turned out to be very useful for implementations on QPACE, the no 1 top green 500 special purpose computer developed tiogethe with IBM  by the SFB-TR 55 in Regensburg and Wuppertal, and on first results on adaptive approaches for obtaining an appropriate coarse system. 
Martin J. Gander (Universite de Geneve) Why it is so difficult to solve Helmholtz problems with iterative methods
Abstract:

In contrast to the positive definite Helmholtz equation, the
deceivingly similar looking indefinite Helmholtz equation is difficult
to solve using classical iterative methods. Applying directly a Krylov
method to the discretized equations without preconditioning leads in
general to stagnation and very large iteration counts. Using classical
incomplete LU preconditioners can even make the situation worse.
Classical domain decomposition and multigrid methods also fail to
converge when applied to such systems.

The purpose of this presentation is to investigate in each case where
the problems lie, and to explain why classical iterative methods have
such difficulties to solve indefinite Helmholtz problems. I will also
present remedies that have been proposed over the last decade, for
incomplete LU type preconditioners, domain decomposition and also
multigrid methods.

 

Joscha Gedicke (Humboldt-Universität) Adaptive solution of parametric eigenvalue problems for partial differential equations
Abstract: Eigenvalue problems for partial differential equations (PDEs) arise in a large number of current technological applications, e.g., in the computation of the acoustic field inside vehicles (such as cars, trains or airplanes). Another current key application is the noise compensation in highly efficient motors and turbines. For the analysis of standard adaptive finite element methods an exact solution of the discretized algebraic eigenvalue problem is required, and the error and complexity of the algebraic eigenvalue problems are ignored. In the context of eigenvalue problems these costs often dominate the overall costs and because of that, the error estimates for the solution of the algebraic eigenvalue problem with an iterative method have to be included in the adaptation process. The goal of our work is to derive adaptive methods of optimal complexity for the solution of PDE-eigenvalue problems including problems with parameter variations in the context of homotopy methods. In order to obtain low (or even optimal) complexity methods, we derive and analyse methods that adapt with respect to the computational grid, the accuracy of the iterative solver for the algebraic eigenvalue problem, and also with respect to the parameter variation. Such adaptive methods require the investigation of a priori and a posteriori error estimates in all three directions of adaptation. As a model problem we study eigenvalue problems that arise in convection-diffusion problems. We developed robust a posteriori error estimators for the discretization as well as for the iterative solver errors, first for self-adjoint second order eigenvalue problems (undamped problem, diffusion problem), and then bring in the non-selfadjoint part (damping, convection) via a homotopy, where the step-size control for the homotopy is included in the adaptation process.
Joscha Gedicke (Humboldt-Universität) Adaptive solution of parametric eigenvalue problems for partial differential equations
Abstract: Eigenvalue problems for partial differential equations (PDEs) arise in a large number of current technological applications, e.g., in the computation of the acoustic field inside vehicles (such as cars, trains or airplanes). Another current key application is the noise compensation in highly efficient motors and turbines. For the analysis of standard adaptive finite element methods an exact solution of the discretized algebraic eigenvalue problem is required, and the error and complexity of the algebraic eigenvalue problems are ignored. In the context of eigenvalue problems these costs often dominate the overall costs and because of that, the error estimates for the solution of the algebraic eigenvalue problem with an iterative method have to be included in the adaptation process. The goal of our work is to derive adaptive methods of optimal complexity for the solution of PDE-eigenvalue problems including problems with parameter variations in the context of homotopy methods. In order to obtain low (or even optimal) complexity methods, we derive and analyse methods that adapt with respect to the computational grid, the accuracy of the iterative solver for the algebraic eigenvalue problem, and also with respect to the parameter variation. Such adaptive methods require the investigation of a priori and a posteriori error estimates in all three directions of adaptation. As a model problem we study eigenvalue problems that arise in convection-diffusion problems. We developed robust a posteriori error estimators for the discretization as well as for the iterative solver errors, first for self-adjoint second order eigenvalue problems (undamped problem, diffusion problem), and then bring in the non-selfadjoint part (damping, convection) via a homotopy, where the step-size control for the homotopy is included in the adaptation process.
Joscha Gedicke (Humboldt-Universität) Adaptivity for the Hodge decomposition of Maxwell's equations
Abstract: Recently a new numerical method for the two-dimensional Maxwell's equation based on the Hodge decomposition for divergence-free vector fields has been introduced by Brenner, Cui, Nan and Sung. The advantage of this new approach is that an approximation of the vector field is obtained by solving several standard second order scalar elliptic boundary value problems instead of using more complicated methods. For the linear Courant finite elements standard energy residual type a posteriori error techniques can be applied to obtain guaranteed upper bounds for the L2-error. For smooth solutions a duality argument shows reliability of an L2 residual type a posteriori error estimator for the H(curl)-error. A dual weighted residual error estimator is derived for singular solutions. Numerical experiments verify reliability and show empirically efficiency of the proposed error estimators. It is shown that adaptive mesh-refinement numerically leads to optimal convergence rates for general domains that are non-convex and may include holes.
Marc Iwan Gerritsma (Technische Universiteit te Delft) Cochain interpolation for spectral element methods
Abstract: Cochains are the natural discrete analogues of the continuous differential forms. The exterior derivative is replaced in the discrete setting by the coboundary operator. In this way the vector operations grad, curl and div are encoded. Since application of the coboundary twice yields the zero operator, the vector identities div curl = 0 and curl grad = 0 are identically satisfied on arbitrarily shaped grids, since the coboundary operator acts on cochains in a purely topological sense.

For the application of cochains in numerical methods cochain interpolation is required which needs to satisfy two criteria:

1. When the interpolated k-cochain is integrated over a k-chain, the cochain should be retrieved.

2. The interpolated k-cochain should be close the corresponding continuous k-form in some norm.

In this poster cochain interpolations will be presented which satisfy criterion 1. and which display exponential convergence with polynomial enrichment for suffiently smooth k-forms. Several examples of the use of these interpolating functions will be presented, such as:

1. Discrete conservation laws naturally reduce to finite volume discretizations.

2. The condition number of the resulting system matrix grows much slower with polynomial enrichment than conventional spectral methods.

3. Low order finite volume methods are extremely good preconditioners.

4. The resonant cavity eigenvalue problem in a square box is resolved with exponential accuracy on orthogonal and highly deformed grids, whereas conventional spectral methods fail to do so.

References:

[1] Marc Gerritsma, Edge functions for spectral element methods, Proceedings of ICOSAHOM 2009, 2010.

[2] Mick Bouman, Artur Palha, Jasper Kreeft and Marc Gerritsma, A conservative spectral element method for curvilinear domains, Proceedings of ICOSAHOM2009, 2010.

[3] Bochev, P.B. and J.M. Hyman, Principles of mimetic discretizations of differential operators, IMA Volumes In Mathematics and its Applications, 142, 2006.
Andrew Kruse Gillette (University of Texas at Austin) Error estimates for generalized barycentric interpolation
Abstract: We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions. This is joint work with Dr. Alexander Rand and Dr. Chandrajit Bajaj.
Shiyuan Gu (Louisiana State University) A C0 interior penalty method for a biharmonic problem with essential and natural boundary conditions of Cahn-Hilliard type
Abstract: We develop a C0 interior penalty method for a biharmonic problem with essential and natural boundary conditions of Cahn-Hilliard type. Both a priori and a posteriori error estimates are derived. C0 interior penalty methods are much simpler than C1 finite element methods. Compared to mixed finite element methods, the stability of C0 interior penalty methods can be established in a straightforward manner and the symmetric positive definiteness of the continuous problems is preserved by C0 interior penalty methods. Furthermore, since the underlying finite element spaces are standard spaces for second order problems, multigrid solves for the Laplace operator can be used as natural preconditioners for C0 interior penalty methods.
Ronald H.W. Hoppe (University of Houston) Projection based model reduction for shape optimization of the Stokes system
Abstract:

The optimal design of structures and systems described by partial differential equations (PDEs) often gives rise to large-scale optimization problems, in particular if the underlying system of PDEs represents a multi-scale, multi-physics problem. Therefore, reduced order modeling techniques such as balanced truncation model reduction, proper orthogonal decomposition, or reduced basis methods are used to significantly decrease the computational complexity while maintaining the desired accuracy of the approximation. In particular, we are interested in such shape optimization problems where the design issue is restricted to a relatively small area of the computational domain. In this case, it appears to be natural to rely on a full order model only in that specific part of the domain and to use a reduced order model elsewhere. An appropriate methodology to realize this idea consists in a combination of domain decomposition techniques and balanced truncation model reduction. We will consider such an approach for shape optimization problems associated with the time-dependent Stokes system and derive explicit error bounds for the modeling error.
As an application in life sciences, we will be concerned with the optimal design of surface acoustic wave driven microfluidic biochips that are used in clinical diagnostics, pharmacology, and forensics for high-throughput screening and hybridization in genomics and protein profiling in proteomics.

Ronald H.W. Hoppe (University of Houston) An introduction to the a posteriori error analysis of elliptic optimal control problems
Abstract: We give a survey on adaptive finite element methods for optimal control problems associated with second order elliptic boundary value problems. Both unconstrained and constrained problems will be considered, the latter in case of pointwise control and pointwise state constraints. Mesh adaptivity is realized in terms of a posteriori error estimators obtained by using residual-type error control and/or by weighted dual residuals within the goal-oriented dual weighted approach. In order to make the exposition self-contained, we provide the basic concepts of residual-type and goal-oriented a posteriori error control for elliptic boundary value problems and then apply both concepts to unconstrained elliptic optimal control problems. Control constrained problems will be exemplarily treated within the residual-type a posteriori error analysis, whereas the case of pointwise state constraints will be dealt with by means of dual weighted residuals.

The results are based on joint work with Michael Hintermueller (Humboldt Univ. at Berlin) and Michael Hinze (Univ. of Hamburg).
Thomas Yizhao Hou (California Institute of Technology) Adaptive data analysis via nonlinear compressed sensing
Abstract: We introduce a new adaptive data analysis method to analyze multiscale nonlinear and non-stationary data. The purpose of this work is to find the sparsest representation of a multiscale signal using basis that is adapted to the data instead of being prescribed a priori. Using a variation approach base on nonlinear L1 optimization, our method defines trends and Instantaneous Frequency of amultiscale signal. One advantage of performing such decomposition is to preserve some intrinsic physical property of the signal without introducing artificial scales or harminics. For multiscale data that have a nonlinear sparse representation, we prove that our nonlinear optimization method converges to the exact signal with the sparse representation. Moreover, we will show that our method is insensitive to noise and robust enough to apply to realistic physical data. For general data that do not have a sparse representation, our method will give an approximate decomposition and the accuracy is controlled by the scale separation property of the original signal.
Chiu-Yen Kao (Ohio State University) An efficient rearrangement algorithm for shape optimization on eigenvalue problems
Abstract: In this poster, an efficient rearrangement algorithm is proposed to find the optimal shape and topology for eigenvalue problems in an inhomogeneous media. The method is based on Rayleigh quotient formulation of eigenvalue and a monotone iteration process to achieve the optimality. The common numerical approach for these problems is to start with an initial guess for the shape and then gradually evolve it, until it morphs into the optimal shape. One of the difficulties is that the topology of the optimal shape is unknown. Developing numerical techniques which can automatically handle topology changes becomes essential for shape and topology optimization problems. The level set approach based on both shape derivatives and topological derivatives has been well known for its ability to handle topology changes. However, CFL constrain significantly slows down the algorithm when the mesh is further refined. Due to the efficient rearrangement, the new method not only has the ability of topological changes but also is exempt from CFL condition. We provides numerous numerical examples to demonstrate the robustness and efficiency of our approach.
David Keyes (King Abdullah University of Science & Technology) Domain decomposition methods for partial differential equations
Abstract: Domain decomposition, a form of divide-and-conquer for mathematical problems posed over a physical domain is the most common paradigm for large-scale simulation on massively parallel, distributed, hierarchical memory computers. In domain decomposition, a large problem is reduced to a collection of smaller problems, each of which is easier to solve computationally than the undecomposed problem, and most or all of which can be solved independently and concurrently. Domain decomposition has proved to be an ideal paradigm not only for execution on advanced architecture computers, but also for the development of reusable, portable software. The most complex operation in a typical domain decomposition method – the application of the preconditioner – carries out in each subdomain steps nearly identical to those required to apply a conventional preconditioner to the undecomposed domain. Hence software developed for the global problem can readily be adapted to the local problem, instantly presenting lots of legacy scientific code for to be harvested for parallel implementations. Finally, it should be noted that domain decomposition is often a natural paradigm for the modeling community. Physical systems are often decomposed into two or more contiguous subdomains based on phenomenological considerations, and the subdomains are discretized accordingly, as independent tasks. This physically-based domain decomposition may be mirrored in the software engineering of the corresponding code, and leads to threads of execution that operate on contiguous subdomain blocks. This tutorial provides an overview of domain decomposition and focuses on the mathematical development of its two main paradigms: Schwarz and Schur preconditioning and their hybrids.
David Keyes (King Abdullah University of Science & Technology) Preconditioners for interface problems in Eulerian formulations
Abstract: Eulerian formulations of problems with interfaces avoid the subtleties of tracking and remeshing, but do they complicate solution of the discrete equations, relative to domain decomposition methods that respect the interface? We consider two different interface problems – one involving cracking and one involving phase separation. Crack problems can be formulated by extended finite element methods (XFEM), in which discontinuous fields are represented via special degrees of freedom. These DOFs are not properly handled in a typical AMG coarsening process, which leads to slow convergence. We propose a Schwarz approach that retains AMG advantages on the standard DOFs and avoids coarsening the enriched DOFs. This strategy allows reasonably mesh-independent convergence rates, though the convergence degradation of the (lower dimensional) set of crack DOFs remains to be addressed. Phase separation problems can be formulated by the Cahn-Hilliard approach, in which the level set of a continuous Eulerian field demarcates the phases. Here, scalable preconditioners follow naturally, once the subtlety of the temporal discretization is sorted out. The first project is joint with R. Tuminaro and H. Waisman and the second with X.-C. Cai and C. Yang.

Jon Kleinberg (Cornell University) Arnold family lecture: Burst, cascades, and hot spots: A glimpse of some on-line social phenomena at global scales
Abstract: As an increasing amount of social interaction moves on-line, it becomes possible to study phenomena that were once essentially invisible: how our social networks are organized, how groups of people come together and attract new members, and how information spreads through society. With computational and mathematical ideas, we can begin to map the rich social landscape that emerges, filled with "hot spots" of collective attention, and behaviors that cascade through our networks of social connections.
Tzanio V Kolev (Lawrence Livermore National Laboratory) Scalable electromagnetic simulations with the Auxiliary-space Maxwell Solver (AMS)
Abstract: Second-order definite Maxwell problems arise in many practical applications, such as the modeling of electromagnetic diffusion in ALE-MHD simulations. Typically, these problems are discretized with Nedelec finite elements resulting in a large sparse linear system which is challenging for linear solvers due to the large nullspace of curl-operator. In this poster we describe our work on the Auxiliary-space Maxwell Solver (AMS) which is a provably efficient scalable code for solving definite Maxwell problems based on the recent Hiptmair-Xu (HX) decomposition of the lowest-order Nedelec space. We demonstrate the scalability of the method and its robustness with respect to jumps in material coefficients. We also report some results from recent work on the algebraic extension of the AMS algorithm and theory to linear systems obtained by explicit element reduction.
Ralf Kornhuber (Freie Universität Berlin) Nonsmooth Schur Newton Methods and Applications
Abstract:
The numerical simulation of the coarsening of binary alloys based on the
Cahn-Larch`e equations requires fast, reliable and robust solvers for Cahn-Hilliard equations
with logarithmic potential. After semi-implicit time discretization (cf.  Blowey and Elliott 92),
the resulting spatial problem can be reformulated as a non-smooth pde-constrained
optimal control problem with cost functional induced by the Laplacian. The associated
Karush-Kuhn-Tucker conditions take the form of a nonsmooth saddle point problem
degenerating to a variational inclusion in the deep quench limit.
Our considerations are based on recent work of Gr¨aser & Kornhuber 09 and the
upcoming dissertation of Gr¨aser 10. The starting point is the elimination of the
primal variable leading to a nonlinear Schur complement which turns out to be the
Fr´ech`et derivative of a convex functional. Now so-called nonsmooth Schur-Newton
methods can be derived as gradient-related descent methods applied to this functional.
In the discrete case we can show global convergence for an exact and
an inexact version independent of any regularization parameters. Local quadratic
convergence or finite termination can be shown for piecewise smooth nonlinearities
or in the deep quench limit respectively. The algorithm can be reinterpreted as a
preconditioned Uzawa method and generalizes the well-known primal-dual active
set strategy by Kunisch, Ito, and Hinterm¨uller 03. A (discrete) Allen-Cahn-type
problem and a linear saddle point problem have to be solved (approximately) in
each iteration step. In numerical computations we observe mesh-independent local
convergence for initial iterates provided by nested iteration. In the deep quench
limit, the numerical complexity of the (approximate) solution of the arising linear
saddle point problem dominates the detection of the actual active set.
Christian Kreuzer (Universität Duisburg-Essen) Convergence and optimality of adaptive finite element methods
Abstract: We present convergence and optimality results for a standard AFEM
    SOLVE -> ESTIMATE -> MARK -> REFINE.
The results range from plain convergence for inf-sup problems to contraction properties and quasi-optimal rates of an AFEM with D"orfler marking for elliptic problems. Beyond that we show how these results can be used to design convergent adaptive methods for a linear parabolic pde and a nonlinear stationary Stokes problem.
Angela Kunoth (Universität Paderborn) Space-time adaptive wavelet methods for control problems constrained by parabolic PDEs
Abstract: Joint work with Max D. Gunzburger, School of Computational Science, Florida State University.

Optimization problems constrained by partial differential equations (PDEs) are particularly challenging from a computational point of view: the first order necessary conditions for optimality lead to a coupled system of PDEs. Specifically, for the solution of control problems constrained by a parabolic PDE, one needs to solve a system of PDEs coupled globally in time and space. For these, conventional time-stepping methods quickly reach their limitations due to the enormous demand for storage. For such a coupled PDE system, adaptive methods which aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities in the data or domain, with respect to both space and time, appear to be most promising.

Here I propose an adaptive method based on wavelets. It builds on a recent paper by Schwab and Stevenson where a single linear parabolic evolution problem is formulated in a weak space-time form and where an adaptive wavelet method is designed for which convergence and optimal convergence rates (when compared to wavelet-best N term approximation) can be shown. Our approach extends this paradigm to control problems constrained by evolutionary PDEs for which we can prove convergence and optimal rates for each of the involved unknowns (state, costate, and control).
Angela Kunoth (Universität Paderborn) Multiscale methods for complex systems
Abstract: This poster presents different topics concerning the modelling and numerical solution of complex systems from my work group, all centering around multiscale methods for partial differential equations. Applications are from theoretical physics, geodesy, electrical engineering, and finance. Depending on the concrete application, we employ wavelet, adaptive wavelet or monotone multigrid methods.
Sabine Le Borne (Tennessee Technological University) H-LU factorization of stabilized saddle point problems
Abstract: The (mixed finite element) discretization of the linearized Navier-Stokes equations leads to a linear system of equations of saddle point type. The iterative solution of this linear system requires the construction of suitable preconditioners, especially in the case of high Reynolds numbers. In the past, a stabilizing approach has been suggested which does not change the exact solution but influences the accuracy of the discrete solution as well as the effectiveness of iterative solvers. This stabilization technique can be performed on the continuous side before the discretization, where it is known as ``grad-div'' stabilization, as well as on the discrete side where it is known as an ``augmented Lagrangian'' technique (and does not change the discrete solution).

We study the applicability of H-LU factorizations to solve the arising subproblems in the different variants of stabilized saddle point systems.
Jungho Lee (Argonne National Laboratory) A comparison of two domain decomposition methods for a linearized contact problem
Abstract: We compare two domain decomposition methods for a linearized contact problem. The first method we consider has been used in an engineering community; we provide theoretical and numerical evidence that this method is not scalable with respect to the number of subdomains (processors). We propose a scalable alternative and analyze its properties, both theoretically and numerically. We also solve a model problem using a combination of a primal-dual active set method, viewed as a semismooth Newton method, and the scalable alternative.
Dmitriy Leykekhman (University of Connecticut) Local properties of finite element solutions for advection-dominated optimal control problems
Abstract: We analyzes the local properties of several stabilized methods, namely symmetric interior penalty upwind discontinuous Galerkin method (SIPG) and Streamline diffusion method (SUPG) for the numerical solution of optimal control problems governed by linear reaction-advection-diffusion equations with distributed controls. The theoretical and numerical results presented show that for advection-dominated problems the local convergence properties of the SIPG discretization can be superior to the convergence properties of stabilized finite element discretizations such as SUPG method. For example for a small diffusion parameter the SIPG method is optimal in the interior of the domain. This is in sharp contrast to SUPG discretizations, for which it is known that the existence of boundary layers can pollute the numerical solution of optimal control problems everywhere even into domains where the solution is smooth and, as a consequence, in general reduces the convergence rates to only first order. This favorable property of the SIPG method is due to the weak treatment of boundary conditions which is natural for discontinuous Galerkin methods, while for SUPG methods strong imposition of boundary conditions is more conventional. Our numerical results support this conclusion.
Fengyan Li (Rensselaer Polytechnic Institute) Central DG methods for Hamilton-Jacobi equations and ideal MHD equations
Abstract: In this talk, I will present our recent work in developing high order central discontinuous Galerkin (DG) methods for Hamilton-Jacobi (H-J) equations and ideal MHD equations. Originally introduced for hyperbolic conservation laws, central DG methods combine ideas in central schemes and DG methods. They avoid the use of exact or approximate Riemann solvers, while evolving two copies of approximating solutions on overlapping meshes.

To devise Galerkin type methods for H-J equations, the main difficulty is that these equations in general are not in the divergence form. By recognizing a weighted-residual or stabilization-based formulation of central DG methods when applied to hyperbolic conservation laws, we propose a central DG method for H-J equations. Though the stability and the error estimate are established only for linear cases, the accuracy and reliability of the method in approximating the viscosity solutions are demonstrated through general numerical examples. This work is jointly done with Sergey Yakovlev.

In the second part of the talk, we introduce a family of central DG methods for ideal MHD equations which provide the exactly divergence-free magnetic field. Ideal MHD system consists of a set of nonlinear hyperbolic equations, with a divergence-free constraint on the magnetic field. Though such constraint holds for the exact solution as long as it does initially, neglecting this condition numerically may lead to nonphysical features of approximating solutions or numerical instability. This work is jointly done with Liwei Xu and Sergey Yakovlev.
Guang Lin (Pacific Northwest National Laboratory) Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws
Abstract: We develop a new hierarchical reconstruction (HR) method for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes.
Alexei Lozinski (Université de Toulouse III (Paul Sabatier)) Multiscale finite element method in perforated domains
Abstract: We present an adaptation of a Multiscale Finite Element Method (MsFEM by T. Hou et al.) to a simplified context of pollution dissemination in urban area in a real time marching simulation code. To avoid the use of a complex unstructured mesh that perfectly fits any building of the urban area a penalization technique is used. The physical model becomes a diffusion+penalization equation with highly heterogeneous and discontinuous coefficients. MsFEM is adapted by developing a new basis function oversampling technique. This is tested on a genuine urban area. We also present new variants of MsFEM inspired by the non conforming finite elements à la Crouzeix-Raviart.
Yvon Jean Maday (Université de Paris VI (Pierre et Marie Curie)) Two grids approximation of non linear eigenvalue problems
Abstract:
Approximation of non linear eigenvalue problems represent the key ingredient in quantum chemistry. These approximation are much computer demanding and these approximations saturate the ressources of many HPC centers. Being nonlinear, the approximation methods are iterative and a way to reduce the cost is to use different grids as has been proposed in fluid mechanics for various non linear problems as the Navier Stokes problem. We explain the basics of the approximation, present the numerical analysis and numerical results that  illustrate the good behavior of the two grids scheme. This work has been done in collaboration with Eric Cancès and Rachida Chakir.
 
Niall Madden (National University of Ireland, Galway) Computational aspects of a two-scale finite element method for singularly perturbed problems
Abstract: We consider the numerical solution linear, two dimensional singularly perturbed reaction-diffusion problem posed on a unit square with homogeneous Dirichlet boundary conditions. In [1], it is shown that a two-scale sparse grid finite element method applied to this problem achieves the same order of accuracy as a standard Galerkin finite element method, while reducing the number of degrees of freedom from O(N2) to O(N3/2).

In this presentation, we discuss implementation aspects of the algorithm, particularly regarding the computational cost. We also compare the method with the related "combination" technique.

[1] F. Liu, N. Madden, M. Stynes & A. Zhou, A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions, IMA J. Numer. Anal. 29 (2009), 986-1007.
Niall Madden (National University of Ireland, Galway) Robust numerical solution of singularly perturbed problems
Abstract: Singularly perturbed differential equations are usually posed with a small positive (perturbation) parameter multiplying the highest derivative. Their solutions typically exhibit boundary or interior layers. In recent years much effort has been directed towards constructing and analysing so-called "parameter robust" methods. Such methods should yield solutions whose accuracy does not depend on the perturbation parameter, and should resolve any layers present.

In this talk I will survey some of these methods, and the mathematics behind them, with particular emphasis on finite differences for coupled systems.
Jan Mandel (University of Colorado) Coupled atmosphere - wildland fire numerical simulation by WRF-Fire
Abstract: WRF-Fire consists of a fire-spread model, implemented by the level set method, coupled with the Weather Research and Forecasting model (WRF). In every time step, the fire model inputs the surface wind and outputs the heat flux from the fire. The level set method allows submesh representation of the burning region and flexible implementation of various ignition modes. This presentation will address the numerical methods in the fire module, solving the Hamilton-Jacobi level set equation, modeling real wildfire experiments, visualization, and experimental data assimilation with spatial displacement and representation of smooth random fields by FFT.

Visualizations by Bedrich Sousedik, Erik Anderson, and Joel Daniels.

Jan Mandel, Jonathan D. Beezley, Janice L. Coen, and Minjeong Kim, Data Assimilation for Wildland Fires: Ensemble Kalman filters in coupled atmosphere-surface models, IEEE Control Systems Magazine 29, Issue 3, June 2009, 47-65

Jan Mandel, Jonathan D. Beezley, and Volodymyr Y. Kondratenko, Fast Fourier Transform Ensemble Kalman Filter with Application to a Coupled Atmosphere-Wildland Fire Model. Anna M. Gil-Lafuente, Jose M. Merigo (Eds.) Computational Intelligence in Business and Economics (Proceedings of the MS'10 International Conference, Barcelona, Spain, 15-17 July 2010), World Scientific, pp. 777-784. Also available at arXiv:1001.1588
Thomas A. Manteuffel (University of Colorado) A Parallel, Adaptive, First-Order System Least-Squares (FOSLS) Algorithm for Incompressible, Resistive Magnetohydrodynamics
Abstract:
Magnetohydrodynamics (MHD) is a fluid theory that describes Plasma Physics by treating the plasma as a fluid of charged particles.  Hence, the equations that describe the plasma form a nonlinear system that couples Navier-Stokes with Maxwell's equations. We describe how the FOSLS method can be applied to incompressible resistive MHD to yield a well-posed, H$^1$-equivalent functional minimization.

To solve this system of PDEs, a nested-iteration-Newton-FOSLS-AMG-LAR approach is taken. Much of the work is done on relatively coarse grids, including most of the linearizations.  We show that at most one Newton step and a few V-cycles are all that is needed on the finest grid. Estimates of the local error and of relevant problem parameters that are established while ascending through the sequence of nested grids are used to direct local adaptive mesh refinement (LAR), with the goal of obtaining an optimal grid at a minimal computational cost. An algebraic multigrid solver is used to solve the linearization steps.

A parallel implementation is described that uses a binning strategy. We show that once the solution is sufficiently resolved, refinement becomes uniform which essentially eliminates load balancing on the finest grids.

The ultimate goal is to resolve as much physics as possible with the least amount of computational work. We show that this is achieved in the equivalent of a few dozen work units on the finest grid. (A work unit equals a fine grid residual evaluation).

Numerical results are presented for two instabilities in a large aspect-ratio tokamak, the tearing mode and the island coalescence mode.

 
Jens Markus Melenk (Technische Universität Wien) Wavenumber-explicit convergence analysis for the Helmholtz equation: hp-FEM and hp-BEM
Abstract: We consider boundary value problems for the Helmholtz equation at large wave numbers k. In order to understand how the wave number k affects the convergence properties of discretizations of such problems, we develop a regularity theory for the Helmholtz equation that is explicit in k. At the heart of our analysis is the decomposition of solutions into two components: the first component is an analytic, but highly oscillatory function and the second one has finite regularity but features wavenumber-independent bounds.

This understanding of the solution structure opens the door to the analysis of discretizations of the Helmholtz equation that are explicit in their dependence on the wavenumber k. As a first example, we show for a conforming high order finite element method that quasi-optimality is guaranteed if (a) the approximation order p is selected as p = O(log k) and (b) the mesh size h is such that kh/p is small. As a second example, we consider combined field boundary integral equation arising in acoustic scattering. Also for this example, the same scale resolution conditions as in the high order finite element case suffice to ensure quasi-optimality of the Galekrin discretization.

This work presented is joint work with Stefan Sauter (Zurich) and Maike Löhndorf (Vienna).
Peter Monk (University of Delaware) The solution of time harmonic wave equations using complete families of elementary solutions
Abstract: This presentation is devoted to plane wave methods for approximating the time-harmonic wave equation paying particular attention to the Ultra Weak Variational Formulation (UWVF). This method is essentially an upwind Discontinuous Galerkin (DG) method in which the approximating basis functions are special traces of solutions of the underlying wave equation. In the classical UWVF, due to Cessenat and Després, sums of plane wave solutions are used element by element to approximate the global solution. For these basis functions, convergence analysis and considerable computational experience shows that, under mesh refinement, the method exhibits a high order of convergence depending on the number of plane wave used on each element. Convergence can also be achieved by increasing the number of basis functions on a fixed mesh (or a combination of the two strategies). However ill-conditioning arising from the plane wave basis can ultimately destroy convergence. This is particularly a problem near a reentrant corner where we expect to need to refine the mesh.

The presentation will start with a summary of the UWVF and some typical analytical and numerical results for the Hemholtz equation. An alternative to plane waves, is to use polynomial basis functions on small elements. Using mixed finite element methods, we can view the UWVF as a hybridization strategy and I shall also present theoretical and numerical results for this approach.

Although neither the Bessel function or the plane wave UWVF are free of dispersion error (pollution error) they can provide a method that can use large elements and small number of degrees of freedom per wavelength to approximate the solution. It has been extended to Maxwell's equations and elasticity. Perhaps the main open problems are how to improve on the bi-conjugate gradient method that is currently used to solve the linear system, and how to adaptively refine the approximation scheme.
Brian Edward Moore (University of Central Florida) Conformal conservation laws and geometric integration for Hamiltonian PDE with added dissipation
Abstract: Conformal conservation laws are defined and derived for a class of multi-symplectic equations with added dissipation. In particular, the conservation laws of symplecticity, energy and momentum are considered, along with others that arise from linear symmetries. Numerical methods that preserved these conformal conservation laws are presented. The nonlinear Schrödinger equation and semi-linear wave equation with added dissipation are used as examples to demonstrate the results.
Michael Joseph Neilan (Louisiana State University) Finite element methods for the Monge-Ampere equation
Abstract: The Monge-Ampere equation is a fully nonlinear second order PDE that arises in various application areas such as differential geometry, meteorology, reflector design, economics, and optimal transport. Despite its prevalence in many applications, numerical methods for the Monge-Ampere equation are still in their infancy. In this work, I will discuss a new approach to construct and analyze several finite element methods for the Monge-Ampere equation. As a first step, I will show that a key feature in developing convergent discretizations is to construct schemes with stable linearizations. I will then describe a methodology for constructing finite elements that inherits this trait and provide two examples: C^0 finite element methods and discontinuous Galerkin methods. Finally, I will present some promising application areas to apply this methodology including mesh generation and computing a manifold with prescribed Gauss curvature.
Ngoc-Cuong Nguyen (Massachusetts Institute of Technology) HDG methods for multiphysics simulation
Abstract: We present a recent development of hybridizable discontinuous Galerkin (HDG) methods for continuum mechanics. The HDG methods inherit the geometric flexibility, high-order accuracy, and multiphysics capability of discontinuous Galerkin (HDG) methods. They also possess several unique features which distinguish themselves from other DG methods: (1) the global unknowns are the numerical traces of the field variables; (2) all the approximate variables converge with the optimal order k+1 for diffusion-dominated problems; (3) in such cases, local postprocessing can be developed to increase the convergence rate to k+2 for the approximation of the field variables; (4) they can deal with non-compatible boundary conditions; (5) they result in a compact matrix system and (6) they are somewhat easier to implement and provide a single code for solving multiphysics problems.
Ngoc-Cuong Nguyen (Massachusetts Institute of Technology) HDG methods for CFD applications
Abstract: We extend hybridizable discontinuous Galerkin (HDG) methods to CFD applications. The HDG methods inherit the geometric flexibility and high-order accuracy of discontinuous Galerkin methods, and offer a significant reduction in the computational cost. In order to capture shocks, we employ an artificial viscosity model based on an extension of existing artificial viscosity methods. In order to integrate the Spalart-Allmaras turbulence model using high-order methods, some modification of the model is necessary. Mesh adaptation based on shock indicator is used to improve shock profiles. Several test cases are presented to illustrate the proposed approach.
Sylvain Nintcheu Fata (Oak Ridge National Laboratory) The generalized fundamental theorem of calculus and its applications to boundary element methods
Abstract: An effective technique which employs only the underlying surface discretization to calculate domain integrals appearing in boundary element methods has been developed. The proposed approach first converts a domain integral with continuous or weakly-singular integrand into a boundary integral. The resulting surface integral is then computed via standard quadrature rules commonly used for boundary elements. This transformation of a domain integral into a boundary counterpart is accomplished through a systematic generalization of the fundamental theorem of calculus to higher dimension. In addition, it is established that the higher-dimensional version of the first fundamental theorem of calculus corresponds to the classical Poincaré lemma.


This research was supported by the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under contract DE-AC05-00OR22725 with UT-Battelle, LLC.

Sylvain Nintcheu Fata (Oak Ridge National Laboratory) 3D boundary integral analysis by a precorrected fast Fourier transform algorithm
Abstract: An acceleration of a Galerkin boundary integral equation (BIE) method for solving the three-dimensional Laplace equation is investigated in the context of the precorrected fast Fourier transform (PFFT) scheme. The PFFT technique is an algorithm for rapid computation of the dense matrix-vector products arising in an iterative solution of discretized integral equations. In the PFFT method, the problem domain is overlaid with a regular Cartesian grid that serves as an auxiliary platform for computation. With the aid of the fast Fourier transform (FFT) procedure, the necessary influence matrices of the discretized problem are rapidly evaluated on the Fourier grid in a sparse manner resulting in a significant reduction in execution time and computer memory requirements.


This research was supported by the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under contract DE-AC05-00OR22725 with UT-Battelle, LLC.
Ricardo H. Nochetto (University of Maryland) Adaptive finite element methods
Abstract: Adaptivity is an essential tool in modern scientific and engineering computation that allows one to optimize the computational effort by locating the degrees of freedom where they are most needed, that is in regions of rapid solution variation. Adaptive finite element methods (AFEM) are the most popular and effective numerical methods to solve elliptic PDE, and are driven by a posteriori error estimators. In this tutorial we will discuss the basic structure of AFEM and its main properties, analyze its convergence (contraction property), and derive convergence rates.
Ricardo H. Nochetto (University of Maryland) Convergence rates of AFEM with H -1 Data
Abstract: In contrast to most of the existing theory of adaptive finite element methods (AFEM), we design an AFEM for -Δ u = f with right hand side f in H -1 instead of L2. This has two important consequences. First we formulate our AFEM in the natural space for f, which is nonlocal. Second, we show that decay rates for the data estimator are dominated by those for the solution u in the energy norm. This allows us to conclude that the performance of AFEM is solely dictated by the approximation class of u.

This is joint work with A. Cohen and R. DeVore.
Eun-Hee Park (Louisiana State University) Two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method
Abstract: The weakly over-penalized symmetric interior penalty (WOPSIP) method was introduced for second order elliptic problems by Brenner et al. in 2008. It belongs to the family of discontinuous Galerkin methods. We will discuss two-level additive Schwarz preconditioners for the WOPSIP method. The key ingredient of the two-level additive Schwarz preconditioner is the construction of the subdomain solvers and the coarse solver. In our approach, we consider different choices of coarse spaces and intergrid transfer operators. It is shown that the condition number estimates previously obtained for classical finite element methods also hold for the WOPSIP method. Numerical results will be presented, which illustrate the parallel performance of these preconditioners.

This is joint work with Andrew T. Barker, Susanne C. Brenner, and Li-yeng Sung.
Ilaria Perugia (Università di Pavia) Trefftz-discontinuous Galerkin methods for time-harmonic wave problems
Abstract:

Several finite element methods used in the numerical discretization of wave problems in frequency domain are based on incorporating a priori knowledge about the differential equation into the local approximation spaces by using Trefftz-type basis functions, namely functions which belong to the kernel of the considered differential operator. For the Helmholtz equation, for instance, examples of Trefftz basis functions are plane waves, Fourier-Bessel functions and Hankel functions, and there are in the literature several methods based on them: the Plane Wave/Bessel Partition of Unit Method by Babuška and Melenk, the Ultra Weak Variational Formulation by Cessenat and Després, the Plane Wave/Bessel Least Square Method by Monk and Wang, the Discontinuous Enrichment Method by Farhat and co-workers, the Method of Fundamental Solutions by Stojek, to give some examples. These methods differ form one another not only for the type of Trefftz basis functions used in the approximating spaces, but also for the way of imposing continuity at the interelement boundaries: partition of unit, least squares, Lagrange multipliers or discontinuous Galerkin techniques. In this talk, the construction of Trefftz-discontinuous Galerkin methods for both the Helmholtz and the time-harmonic Maxwell problems will be reviewed and their abstract error analysis will be presented. It will also be shown how to derive best approximation error estimates for Trefftz functions, needed to complete the convergence analysis, by using Vekua's theory. Some explicit estimates in the case of plane waves will be given. These results have been obtained in collaboration with Ralf Hiptmair and Andrea Moiola form ETH Zürich.

Jingmei Qiu (Colorado School of Mines) High order integral deferred correction method based on Strang split semi-Lagrangian WENO method for Vlasov Poisson simulations
Abstract: We apply the very high order Strang split semi-Lagrangian WENO algorithm for kinetic equations. The spatial accuracy of the current Strang split finite difference WENO algorithm is very high (as high as ninth order), however the temporal error is dominated by the dimensional splitting, which is only second order accurate. It is therefore very important to overcome this splitting error, in order to have a consistently high order numerical algorithm. We are currently working on using the IDC framework to overcome the `at best' second order Strang splitting error. Specifically, the dimensional splitting error is overcomed by iteratively correcting the numerical solution via the error function, which is solved by approximating the error equation. We will show numerically that if one embeds a first order dimensional splitting algorithm into the IDC framework, there will be first order increase in order of accuracy when one applies a correction loop in IDC algorithm. Applications to the Vlasov-Poisson system will be presented.
Gianluigi Rozza (École Polytechnique Fédérale de Lausanne (EPFL)) A reduced basis hybrid method for viscous flows in parametrized complex networks
Abstract: Model order reduction techniques provide an efficient and reliable way of solving partial differential equations in the many-query or real- time context, such as (shape) optimization, characterization, parameter estimation and control.

The reduced basis (RB) approximation is used for a rapid and reliable solution of parametrized partial differential equations (PDEs). The reduced basis method is crucial to find the solution of parametrized problems as projection of previously precomputed solutions for certain instances of the parameters. It consists on rapidly convergent Galerkin approximations on a space spanned by “snapshots” on a parametrically induced solution manifold; rigorous and sharp a posteriori error estimators for the outputs/quantities of interest; efficient selection of quasi-optimal samples in general parameter domains; and Offline-Online computational procedures for rapid calculation in the many-query and real-time contexts.

The error estimators play an important role in efficient and effective sampling procedures: the inexpensive error bounds allow to explore much larger subsets of parameter domain in search of most representative or best “snapshots”, and to determine when we have just enough basis functions.

Extensions of the RB method have been combined with domain decomposition tecniques: this approach, called reduced basis element method (RBEM), is suitable for the approximation of the solution of problems described by partial differential equations within domains which are decomposable into smaller similar blocks and properly coupled. The goal is to speed up the computational time with rapid and efficient numerical strategies to deal with complex and realistic configurations, where topology features are recurrent. The construction of the map from the reference shapes to each corresponding block of the computational domain is done by the generalized transfinite maps. The empirical interpolation procedure has been applied to the geometrical non-affine transformation terms to re- cast the problem in an affine setting.

Domain decomposition techniques are important to enable the use of parallel architectures in order to speed up the computational time, compared to a global approach, and also to increase the geometric complexity dealing with independent smaller tasks on each sub-domain, where the approximated solution is recovered as projection of local previously computed solutions and then properly glued through different domains by some imposed coupling conditions to guarantee the continuity of stresses and velocities in viscous flows, for example. The Offline/Online decoupling of the reduced basis procedure and the computational decomposition of the method allow to reduce considerably the problem complexity and the simulation times.

We propose here an option for RBEM, called reduced basis hybrid method (RBHM) where we focus on different coupling conditions to guarantee the continuity of velocity and pressure. Each basis function in each reference subdomain is computed considering zero- stress condition at the interfaces, the continuity of the stresses (non-zero) at the interfaces is recovered by a coarse finite element solution on the global domain, while the continuity of velocities is guaranteed by Lagrange multipliers. This computational procedure allows to reduce considerably the problem complexity and the computational times which are dominated online by the coarse finite element solution, while all the RB offline calculations may be carried out by a parallel computing approach. Applications and results are shown on several combinations of geometries representing cardiovascular networks made up of stenosis, bifurcation, ect.

Ulrich Rüde (Friedrich-Alexander-Universität Erlangen-Nürnberg) Towards Exascale Computing: Multilevel Methods and Flow Solvers for Millions of Cores
Abstract:
We will report on our experiences implementing PDE solvers on Peta-Scale computers, such as the 290 000 core IBM Blue Gene system in the Jülich Supercomputing Center. The talk will have two parts, the first one reporting on our Hierarchical Hybrid Grid  method, a prototype Finite Element Multigrid Solver scaling up to a trillion (10^12)  degrees of freedom on a tetrahedral mesh by using a carefully designed matrix-free implementation.  The second part of the talk will present our work on simulating complex flow phenomena using the Lattice-Boltzmann algorithm. Our software includes parallel algorithms for treating free surfaces with the goal of simulating fully resolved bubbly flows and foams. Additionally, we will report on a parallel fluid-structure-interaction technique with many moving rigid objects. This work is directed towards the modeling of particulate flows that we can represent using fully resolved geometric models of each individual particle embedded in the flow. The talk will end with some remarks on the challenges that algorithm developers will be facing on the path to exascale in the coming decade.

Giancarlo Sangalli (Università di Pavia) Isogeometric Analysis for electromagnetic problems
Abstract: The concept of Isogeometric Analysis (IGA) was first applied to the approximation of Maxwell equations in [A. Buffa, G. Sangalli, R. Vázquez, Isogeometric analysis in electromagnetics: B-splines approximation, CMAME, doi:10.1016/j.cma.2009.12.002.]. The method is based on the construction of suitable B-spline spaces such that they conform a De Rham diagram. Its main advantages are that the geometry is described exactly with few elements, and the computed solutions are smoother than those provided by finite elements. We present here the theoretical background to the approximation of vector fields in IGA. The key point of our analysis is the definition of suitable projectors that render the diagram commutative. The theory is then applied to the numerical approximation of Maxwell source and eigen problem, and numerical results showing the good behavior of the scheme are also presented.
Christoph Schwab (ETH Zürich) Sparse tensor Galerkin discretizations for first order transport problems
Abstract: Joint with R. Hiptmair, Konstantin Grella, Eividn Fonn of SAM, ETH.

We report on an ongoing project on Sparse Tensor Finite Element Discretizations for High Dimensional Linear Transport Problems.

After reviewing several well-posed variational formulations and the regularity of weak solutions of these problems, we discuss their stable discretizations, with a focus on hierarchic, multilevel type discretizations. Particular examples include (multi)wavelet and shearlet discretizations. We discuss sparse tensor discretizations for Least Squares Formulations of first order transport equations on high dimensional parameter spaces. The formulation is due to Manteuffel etal. (SINUM2000). We present preliminary numerical results for both, sparse tensor spectral as well as for wavelet discretizations.

Results are report from ongoing work at the Seminar for Applied Mathematics at ETH Zurich which is supported by the Swiss National Science Foundation (SNF) and from joint work with the groups of W. Dahmen and of G. Kutyniok within the Priority Research Programme (SPP) No. 1324 of the German Research Foundation. http://www.dfg-spp1324.de
Marc Alexander Schweitzer (Rheinische Friedrich-Wilhelms-Universität Bonn) Stable enrichment and treatment of complex domains in the particle–partition of unity method
Abstract:
We are concerned with the stability and approximation properties of enriched meshfree methods for the discretization of PDE on arbitrary domains. In particular we focus on the particle-partition of unity method (PPUM) yet the presented results hold for any partition of unity based enrichment scheme. The goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an enrichment zone hierarchically near the singularities of the solution. This initial enrichment however can lead to a severe ill-conditioning and can compromise the stability of the discretization. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points. The treatment of general domains with mesh-based methods such as the finite element method is rather involved due to the necessary mesh-generation. In collocation type meshfree methods this complex pre-processing step is completely avoided by construction. However, in Galerkin type meshfree discretization schemes we must compute domain and boundary integrals and thus must be concerned with the meshfree treatment of arbitrary domains. Here, we present a cut-cell-type scheme for the partition of unity method and ensure stability by enforcing the flat-top condition in a simple post-processing step.
Jie Shen (Purdue University) New efficient spectral methods for high-dimensional PDEs and for Fokker-Planck equation of FENE dumbbell model
Abstract: Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality".

We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular, rigorous error estimates as well as efficient numerical algorithms for elliptic equations.

We shall also propose a new weighted weak formulation for the Fokker-Planck equation of FENE dumbbell model, and prove its well-posedness in weighted Sobolev spaces. Based on the new formulation, we are able to design simple, efficient, and unconditionally stable semi-implicit Fourier-Jacobi schemes for the Fokker-Planck equation of FENE dumbbell model.

It is hoped that the combination of the two new approaches would make it possible to directly simulate the five or six dimensional Navier-Stokes Fokker-Planck system.
Chi-Wang Shu (Brown University) Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin and finite volume schemes for conservation laws
Abstract: We construct uniformly high order accurate discontinuous Galerkin (DG) and weighted essentially non-oscillatory (WENO) finite volume (FV) schemes satisfying a strict maximum principle for scalar conservation laws and passive convection in incompressible flows, and positivity preserving for density and pressure for compressible Euler equations. A general framework (for arbitrary order of accuracy) is established to construct a limiter for the DG or FV method with first order Euler forward time discretization solving one dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle and make the scheme uniformly high order in space and time. One remarkable property of this approach is that it is straightforward to extend the method to two and higher dimensions. The same limiter can be shown to preserve the maximum principle for the DG or FV scheme solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. A suitable generalization results in a high order DG or FV scheme satisfying positivity preserving property for density and pressure for compressible Euler equations. Numerical tests demonstrating the good performance of the scheme will be reported. This is a joint work with Xiangxiong Zhang.
Rob Stevenson (Universiteit van Amsterdam) Adaptive tensor product wavelet methods for solving well-posed operator equations
Abstract: In this talk, we give an overview of adaptive wavelet methods for solving operator equations. In particular, we will focus on the following topics: The application of these methods to time evolution problems as parabolic problems and the instationary Stokes system; the advantage of the application of tensor product wavelets and the role of anisotropic regularity; the construction of piecewise tensor product wavelet bases on general domains; the application of the adaptive scheme to singularly perturbed problems.
Tong Sun (Bowling Green State University) Numerical smoothness and error analysis for RKDG on the scalar nonlinear conservation laws
Abstract: The new concept of numerical smoothness is applied to the RKDG (Runge-Kutta/Discontinuous Galerkin) methods for scalar nonlinear conservations laws. The main result is an a posteriori error estimate for the RKDG methods of arbitrary order in space and time, with optimal convergence rate. Currently, the case of smooth solutions is the focus point. However, the error analysis framework is prepared to deal with discontinuous solutions in the future.
Yi Sun (Statistical and Applied Mathematical Sciences Institute (SAMSI)) Numerical study of singular solutions of relativistic Euler equations
Abstract: Singularity formation in relativistic flow is an open theoretical problem in relativistic hydrodynamics. These singularities can be either shock formation, violation of the subluminal conditions or concentration of the mass. We numerically investigate singularity formation in solutions of the relativistic Euler equations in (2+1)-dimensional spacetime with smooth initial data. A hybrid method is used to solve the radially symmetric case. The hybrid method takes the Glimm scheme for an accurate treatment of non-linear waves and a central-upwind scheme in other regions where the fluid flow is sufficiently smooth. The numerical results indicate that shock formation occurs in a certain parametric regime. This is a joint work with Pierre Gremaud.
Vidar Thomée (Chalmers University of Technology) Plenary talk: On the lumped mass finite element method for parabolic problems
Abstract: We study the lumped mass method for the model homogeneous heat equation with homogeneous Dirichlet boundary conditions. We first recall that the maximum principle for the heat equation does not carry over to the the spatially semidiscrete standard Galerkin finite element method, using continuous, piecewise linear approximating functions. However, for the lumped mass variant the situation is more advantageous. We present necessary and sufficient conditions on the triangulation, expressed in terms of properties of the stiffness matrix, for the semidiscrete lumped mass solution operator to be a positive operator or a contraction in the maximum-norm.

We then turn to error estimates in the L2-norm. Improving earlier results we show that known optimal order smooth initial data error estimates for the standard Galerkin method carry over to the lumped mass method, whereas nonsmooth initial data estimates require special assumptions on the triangulations.

We also discuss the application to time discretization by the backward Euler and Crank-Nicolson methods.
Ray S. Tuminaro (Sandia National Laboratories) Energy minimization algebraic multigrid: Robustness and flexibility in multilevel software
Abstract: Energy minimization provides a general framework for developing a family of multigrid algorithms. The proposed strategy is applicable to Hermitian, non-Hermitian, definite, and indefinite problems. Each column of the grid transfer operator P is minimized in an energy-based norm while enforcing two types of constraints: a defined sparsity pattern and preservation of specified modes in the range of P. A Krylov-based strategy is used to minimize energy, which is equivalent to solving A P = 0 with the constraints ensuring a nontrivial solution. For the Hermitian positive definite case, a conjugate gradient-based (CG) method is utilized to construct grid transfers, while methods based on generalized minimum residual (GMRES) and CG on the normal equations (CGNR) are explored for the general case.

One of the main advantages of the approach is that it is flexible, allowing for arbitrary coarsenings, unrestricted sparsity patterns, straightforward long distance interpolation, and general use of constraints, either user-defined or auto-generated. We illustrate how this flexibility can be used to adapt an algebraic multigrid scheme to an extended finite element discretization suitable for modeling fracture. Computational results are presented illustrate that this particular energy minimization scheme gives rise to mesh independent convergence rates and is relatively insensitive to the number and location of cracks being modeled.
Catalin Turc (Case Western Reserve University) Efficient, accurate and rapidly-convergent algorithms for the solution of three dimensional acoustic and electromagnetic scattering problems in domains with geometric singularities
Abstract: We present novel discretization techniques based on boundary integral equations formulations for the solution of three dimensional acoustic and electromagnetic scattering problems in domains with corners and edges. Our method is based on three main components: (1) the use of regularization/preconditioning techniques to design well-conditioned boundary integral equations in domains with geometric singularities; (2) the use of ansatz formulations that explicitly account for the singular and possibly unbounded behavior of the quantities that enter the integral formulations; and (3) the use of a novel Nystrom discretization technique based on non-overlapping integration patches and Chebyshev discretization together with Clenshaw-Curtis-type integrations. We will illustrate the excellent performance of our solvers for a variety of challenging 3D configurations that include closed/open domains with corners and edges. Joint work with O. Bruno (Caltech) and A. Anand (IIT Kanpur).
Andreas Michael Veeser (Università di Milano) Local and global approximation of gradients with piecewise polynomials
Abstract:
The quality of a finite element solution hinges in particular on the approximation properties of the finite element space.  In the first part of this talk we will consider the approximation of the gradient of a target function by continuous piecewise polynomials over a simplicial, 'shape-regular' mesh and prove the following result: the global best approximation error is equivalent to an appropriate sum in terms of the local best approximation errors on the elements, which do not overlap.  This means in particular that, for gradient norms, the continuity requirement does not downgrade the local approximation potential on elements and that discontinuous piecewise polynomials do not offer additional approximation power.  In the second part of the talk we will discuss the usefulness of this result in the context of adaptive methods for partial differential equations. Joint work with Francesco Mora (Milan).
Peter Edward Vincent (Stanford University) The flux reconstruction approach to high-order methods: Theory and application
Abstract: High-order flux reconstruction (FR) schemes are efficient, simple to implement, and allow various high-order methods, such as the nodal discontinuous Galerkin (DG) method and any spectral difference method, to be cast within a single unifying framework. Recently, we have identified a new class of 1D linearly stable FR schemes. Identification of such schemes offers significant insight into why certain FR schemes are stable, whereas other are not. Also, from a practical standpoint, the resulting linearly stable formulation provides a simple prescription for implementing an infinite range of intuitive and apparently robust high-order methods. We are currently extending the 1D formulation to multiple dimensions (including to simplex elements). We are also developing CPU/GPU enabled unstructured high-order inviscid and viscous compressible flow solvers based on the range of linearly stable FR schemes. Details of both the mathematical theory and the practical implementation will be presented in the poster.
Shawn W. Walker (Louisiana State University) Shape optimization of chiral propellers in 3-D stokes flow
Abstract: Locomotion at the micro-scale is important in biology and in industrial applications such as targeted drug delivery and micro-fluidics. We present results on the optimal shape of a rigid body locomoting in 3-D Stokes flow. The actuation consists of applying a fixed moment and constraining the body to only move along the moment axis; this models the effect of an external magnetic torque on an object made of magnetically susceptible material. The shape of the object is parametrized by a 3-D centerline with a given cross-sectional shape. No a priori assumption is made on the centerline. We show there exists a minimizer to the infinite dimensional optimization problem in a suitable infinite class of admissible shapes. We develop a variational (constrained) descent method which is well-posed for the continuous and discrete versions of the problem. Sensitivities of the cost and constraints are computed variationally via shape differential calculus. Computations are accomplished by a boundary integral method to solve the Stokes equations, and a finite element method to obtain descent directions for the optimization algorithm. We show examples of locomotor shapes with and without different fixed payload/cargo shapes.
Wei Wang (Florida International University) High order well-balanced schemes for non-equilibrium flows
Abstract: We studied the well-balancedness properties of the high order finite difference WENO schemes and high order low dissipative filter schemes based on a five-species one-temperature reacting flow model. Both 1d and 2d results are shown to demonstrate the advantages of using well-balanced schemes for non-equilibrium flows.
Timothy C. Warburton (Rice University) GPU accelerated discontinuous Galerkin methods
Abstract: This poster will describe recent progress in adapting discontinuous Galerkin methods to obtain high efficiency on modern graphics processing units. A new low storage version of the methods allows unstructured meshes where all elements to be curvilinear without incurring the usual expensive memory overhead of the traditional scheme. Some performance tests reveal that a modest workstation can generate teraflop performance. Simulation results from time-domain electromagnetics and also compressible flows will demonstrate the promise of this new formulation.
Olof B. Widlund (New York University) An introduction to domain decomposition algorithms
Abstract: Variational formulation and piece-wise linear finite element approximations of Poisson's problem. Dirichlet and Neumann boundary conditions and Poincaré's and Friedrichs's inequalities. A word about linear elasticity. Condition numbers of finite element matrices and the preconditioned conjugate gradient method.

Domains and subdomains. Subdomain matrices as building blocks for domain decomposition methods and the related Schur complements. The two-subdomain case: the Neumann--Dirichlet and Schwarz alternating algorithms; they can be placed in a unified framework and written in terms of Schur complements. Extension to the case of many subdomains; coloring, the problems of singular subdomain matrices, and the need to use a coarse, global problem. Three assumptions and the basic result on the condition number of additive Schwarz algorithms.

Classical and more recent two--level additive Schwarz methods. Remarks on the effect of irregular subdomains. Extensions to elasticity problems including the almost incompressible case.

Modern iterative substructuring methods: FETI–DP and BDDC. An introduction in terms of block-Cholesky for problems only partially assembled. The equivalence of the spectra. Results on elasticity including incompressible Stokes problems.
Olof B. Widlund (New York University) New Domain Decomposition Algorithms from Old
Abstract:
 In recent years, variants of the two-level Schwarz algorithm
have been developed in collaboration between Clark Dohrmann
of Sandia-Albuquerque and a group at the Courant Institute.

By a modification of the coarse component of the preconditioner,
borrowed in part from older domain decomposition methods
of iterative substructuring type, the new methods are easier
to implement for general subdomain geometries and can be made
insensitive to large variations on the coefficients of the partial
differential equation across the interface between the subdomains.

After an introduction to the design of these methods, results on
applications to almost incompressible elasticity and Reissner-Mindlin
plates - solved by using mixed finite elements - and problems
posed in H(div) and H(curl) will be discussed. Some of these results
will appear in the doctoral dissertations of Jong Ho Lee and Duk-soon
Oh, two PhD candidates at the Courant Institute.
Barbara Wohlmuth (Technical University of Munich ) Variational consistent discretization schemes and numerical algorithms for contact problems
Abstract:
 
 
 
We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance.
The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain.
Having a uniform inf-sup bound, one can then establish optimal low-order a priori convergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of the linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations.
Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal-dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual and equilibrated based a posteriori error estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time integration schemes.
However, the differential algebraic character of the system  can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.
 
Carol S. Woodward (Lawrence Livermore National Laboratory) Implicit Solution Approaches: Why We Care and How We Solve the Systems
Abstract:
Parallel computers with large storage capacities have paved the way for increasing both the fidelity and complexity of large-scale simulations.  Increasing fidelity leads to tighter constraints on time steps for stability of explicit methods.  Increasing complexity tends to also increase the number of physics models and variations in time scales.  Providing both a stable solution process which can accurately capture nonlinear coupling between dynamically relevant phenomena while stepping over fast waves or rapid adjustments leads us toward implicit solution approaches.
 
This presentation provides an overview of issues arising in large- scale, multiphysics simulations and some of the motivators for looking at implicit approaches.  We discuss several popular implicit nonlinear solver technologies and show examples of uses of them within the context of problems found in supernova, subsurface simulation, fusion, and nonlinear diffusion problems.
Dexuan Xie (University of Wisconsin) Finite element analysis and a fast solver approach to a nonlocal dielectric continuum model
Abstract: The nonlocal continuum dielectric model is an important extension of the classical Poisson dielectric model. This poster will report some recent results we made on the finite element analysis and fast solver development for one commonly-used nonlocal continuum dielectric model. We first prove that the finite element equation of this model has the unique solution but leads to a dense linear system, which is very expansive to be solved. Surprisingly, we then discover and prove that such a dense linear system can be converted to a system of two sparse finite element equations in a form similar to the standard mixed finite element equation. In this way, fast numerical solvers can be developed to solve the nonlocal continuum dielectric model in an optimal order. Some numerical results in free energy calculation will also be presented to demonstrate the great promise of nonlocal dielectric modeling in improving the accuracy of the classic Poisson dielectric model in computing electrostatic potential energies. This project is a joined work with Prof. Ridgeway Scott, Peter Brune, (both from University of Chicago), and Yi Jiang under the support of NSF grant #DMS-0921004.
Jinchao Xu (Pennsylvania State University) Multilevel iterative methods for PDEs based on one or no grid
Abstract: Several numerical techniques will be presented for solving discretized partial differential equations (PDEs) by special multilevel methods based on one or no grid with nearly optimal computational complexity in a user-friendly fashion.
Liwei Xu (Rensselaer Polytechnic Institute) Numerical simulation of three-dimensional nonlinear water waves
Abstract: We present an accurate and efficient numerical model for the simulation of fully nonlinear (non-breaking), three-dimensional surface water waves on infinite or finite depth. As an extension of the work of Craig and Sulem (1993), the numerical method is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving surface quantities alone. This is accomplished by introducing the Dirichlet--Neumann operator which is described in terms of its Taylor series expansion in homogeneous powers of the surface elevation. Each term in this Taylor series can be computed efficiently using the fast Fourier transform. An important contribution of this paper is the development and implementation of a symplectic implicit scheme for the time integration of the Hamiltonian equations of motion, as well as detailed numerical tests on the convergence of the Dirichlet--Neumann operator. The performance of the model is illustrated by simulating the long-time evolution of two-dimensional steadily progressing waves, as well as the development of three-dimensional (short-crested) nonlinear waves, both in deep and shallow water. This is a joint work with Philippe Guyenne at the University of Delaware.
Liwei Xu (Rensselaer Polytechnic Institute) Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field
Abstract: We present a central discontinuous Galerkin method for solving ideal magnetohydrodynamic (MHD) equations. The methods are based on the original central discontinuous Galerkin methods designed for hyperbolic conservation laws on overlapping meshes, and they use different discretization for magnetic induction equations. The resulting schemes carry many features of standard central discontinuous Galerkin methods such as high order accuracy and being free of exact or approximate Riemann solvers. And more importantly, the numerical magnetic field is exactly divergence-free. Such property, desired in reliable simulations of MHD equations, is achieved by first approximating the normal component of the magnetic field through discretizing induction equations on the mesh skeleton, namely, the element interfaces. And then it is followed by an element-by-element divergence-free reconstruction with the matching accuracy. Numerical examples are presented to demonstrate the high order accuracy and the robustness of the schemes. This is a joint work with Fengyan Li and Sergey Yakovlev at Rensselaer Polytechnic Institute.
Guangri Xue (University of Texas at Austin) A multipoint flux mixed finite element method on general hexahedra: Multiscale mortar extension and applications to multiphase flow in porous media
Abstract: Joint work with Mary Wheeler (University of Texas) and Ivan Yotov (University of Pittsburgh).

We develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell-centered finite difference scheme. A special non-symmetric quadrature rule is employed that yields a positive definite cell-centered system for the scalar by eliminating local fluxes. The method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids, including hexahedra with non-planar faces. Theoretical and numerical results indicate first-order convergence for the scalar and face fluxes. We also develop multiscale mortar method that utilize multipoint flux mixed finite element method as the fine scale discretization. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid scale. With an appropriate choice of polynomial degree of the mortar space, we derive optimal order convergence on the fine scale for both the multiscale pressure and velocity, as well as the coarse scale mortar pressure. We present applications to muliphase flow in porous media.
Ulrike Meier Yang (Lawrence Livermore National Laboratory) A parallel computing tutorial
Abstract: This tutorial will provide an overview of the concepts of parallel computing. Topics to be discussed comprise parallel programming models and computer architectures, including multicore architectures, as well as various issues that need to be considered when designing parallel programs. Several examples of parallel solvers will be presented to illustrate the challenges that need to be overcome to achieve an efficient implementation.
Ulrike Meier Yang (Lawrence Livermore National Laboratory) Hypre: A scalable linear solver library
Abstract: Hypre is a software library for solving large, sparse linear systems of equations on massively parallel computers. The library was created with the primary goal of providing users with advanced parallel preconditioners. The library features parallel multigrid solvers for both structured and unstructured grid problems. For ease of use, these solvers are accessed from the application code via hypre’s conceptual linear system interfaces, which allow a variety of natural problem descriptions. The motivation for the design of hypre, an overview of its interfaces and some of its performance highlights are presented.
Irad Yavneh (Technion-Israel Institute of Technology) Nonlinear Multigrid Revisited
Abstract:

 

Multigrid algorithms for discretized nonlinear partial differential equations and systems are nearly as old as multigrid itself. Over the years several approaches and variants of nonlinear multigrid algorithms have been developed. Typically, for relatively easy problems the different approaches exhibit similar performance. However, for difficult problems the behavior varies, and it is not easy to predict which approach may prevail.

 

In this talk we will consider nonlinear multigrid, focusing on the task of coarse-grid correction, in a general framework of variational coarsening. Such a view reveals clear relations between the various existing approaches and may suggest future variants. This study also sheds light on the choice of inter-grid transfer operators, which are so important for obtaining fast multigrid convergence, and which have received much attention in linear multigrid algorithms but far less so in nonlinear multigrid.     

Shangyou Zhang (University of Delaware) A jumping multigrid method via finite element extrapolation
Abstract: The multigrid method solves the finite element equations in optimal order, i.e., solving a linear system of $O(N)$ equations in $O(N)$ arithmetic operations. Based on low level solutions, we can use finite element extrapolation to obtain the high-level finite element solution on some coarse-level element boundary, at an higher accuracy $O(h_i4)$. Thus, we can solve higher level $(h_j, junderset{sim}{<}2i)$ finite element problems locally on each such coarse-level element. That is, we can skip the finite element problem on middle levels, $h_{i+1},h_{i+2},dots, h_{j-1}$. Roughly speaking, such a jumping multigrid method solves an order $O(N)=O(2^{2di})$ linear system of equations by a memory of $O(sqrt N)=O(2^{di})$, and by a parallel computation of $O(sqrt N)$, where $d$ is the space dimension.

Visitors in Residence
Jeffrey Abell General Motors 11/11/2010 - 11/12/2010
Slimane Adjerid Virginia Polytechnic Institute and State University 10/31/2010 - 11/5/2010
Mark Ainsworth University of Strathclyde 11/28/2010 - 12/4/2010
Alexander Alekseenko California State University 9/1/2010 - 12/31/2010
Roman Andreev ETH Zürich 10/31/2010 - 11/6/2010
Paola Francesca Antonietti Politecnico di Milano 11/27/2010 - 12/3/2010
Todd Arbogast University of Texas at Austin 10/31/2010 - 11/7/2010
Douglas N. Arnold University of Minnesota 9/1/2010 - 6/30/2011
Donald G. Aronson University of Minnesota 9/1/2002 - 8/31/2011
Gerard Michel Awanou Northern Illinois University 9/1/2010 - 6/10/2011
Blanca Ayuso de Dios Centre de Recerca Matemàtica 10/30/2010 - 12/18/2010
Constantin Bacuta University of Delaware 10/31/2010 - 11/7/2010
Nusret Balci University of Minnesota 9/1/2009 - 8/31/2011
Uday Banerjee Syracuse University 9/1/2010 - 12/3/2010
Randolph E. Bank University of California, San Diego 11/28/2010 - 12/2/2010
Andrew T. Barker Louisiana State University 10/31/2010 - 11/6/2010
Andrew T. Barker Louisiana State University 11/27/2010 - 12/3/2010
Yuri Bazilevs University of California, San Diego 10/31/2010 - 11/5/2010
Pavel Belik Augsburg College 11/29/2010 - 12/3/2010
Christine Bernardi Université de Paris VI (Pierre et Marie Curie) 10/31/2010 - 11/5/2010
Pavel B. Bochev Sandia National Laboratories 10/30/2010 - 11/7/2010
Daniele Boffi Università di Pavia 10/30/2010 - 11/7/2010
Andrea Bonito Texas A & M University 11/26/2010 - 12/4/2010
Francesca Bonizzoni Politecnico di Milano 10/15/2010 - 11/10/2010
Carlos Eduardo Cardoso Borges Worcester Polytechnic Institute 11/27/2010 - 12/4/2010
James H. Bramble Texas A & M University 11/5/2010 - 11/7/2010
Susanne C. Brenner Louisiana State University 9/1/2010 - 6/10/2011
Peter Brune University of Chicago 11/27/2010 - 12/6/2010
Xiao-Chuan Cai University of Colorado 11/30/2010 - 12/3/2010
Claudio Canuto Politecnico di Torino 10/17/2010 - 11/7/2010
Varis Carey Colorado State University 11/4/2010 - 11/7/2010
Carsten Carstensen Yonsei University 11/29/2010 - 12/4/2010
Fatih Celiker Wayne State University 9/1/2010 - 12/31/2010
Aycil Cesmelioglu University of Minnesota 9/30/2010 - 8/30/2011
Chi Hin Chan University of Minnesota 9/1/2009 - 8/31/2011
Feng Chen Purdue University 10/30/2010 - 11/4/2010
Long Chen University of California, Irvine 11/28/2010 - 12/3/2010
Qiang Chen University of Delaware 10/31/2010 - 11/6/2010
Yanlai Chen University of Massachusetts, Dartmouth 10/31/2010 - 11/7/2010
Zhiming Chen Chinese Academy of Sciences 10/31/2010 - 11/7/2010
Yingda Cheng Brown University 10/31/2010 - 11/7/2010
Heejun Choi Purdue University 11/28/2010 - 12/3/2010
Shue-Sum Chow Brigham Young University 10/31/2010 - 11/7/2010
Bernardo Cockburn University of Minnesota 9/1/2010 - 6/30/2011
Jintao Cui University of Minnesota 8/31/2010 - 8/30/2011
Wolfgang Dahmen RWTH Aachen 11/1/2010 - 11/6/2010
Christopher Davis University of North Carolina - Charlotte 11/4/2010 - 11/6/2010
Clint Dawson University of Texas at Austin 10/31/2010 - 11/4/2010
Leszek Feliks Demkowicz University of Texas at Austin 10/31/2010 - 11/7/2010
Alan Demlow University of Kentucky 11/2/2010 - 11/6/2010
Alan Demlow University of Kentucky 11/29/2010 - 12/3/2010
Clark R. Dohrmann Sandia National Laboratories 11/29/2010 - 12/3/2010
Tobin A. Driscoll University of Delaware 8/26/2010 - 12/20/2010
Qiang Du Pennsylvania State University 11/2/2010 - 11/7/2010
Yalchin Efendiev Texas A & M University 11/1/2010 - 11/5/2010
Bjorn Engquist University of Texas at Austin 11/1/2010 - 11/5/2010
Randy H. Ewoldt University of Minnesota 9/1/2009 - 8/31/2011
Robert Falgout Lawrence Livermore National Laboratory 11/27/2010 - 12/4/2010
Richard S Falk Rutgers University 9/19/2010 - 12/18/2010
Sean Farley Argonne National Laboratory 11/5/2010 - 11/7/2010
Xiaobing Henry Feng University of Tennessee 10/29/2010 - 12/15/2010
Oscar E. Fernandez University of Minnesota 8/31/2010 - 8/30/2011
Donald A. French University of Cincinnati 11/4/2010 - 11/7/2010
Andreas Frommer Bergische Universität-Gesamthochschule Wuppertal (BUGH) 11/28/2010 - 12/5/2010
Martin J. Gander Universite de Geneve 11/28/2010 - 12/3/2010
Xinfeng Gao Lawrence Berkeley National Laboratory 11/28/2010 - 12/3/2010
Carlos Andres Garavito-Garzon University of Minnesota 11/28/2010 - 11/29/2010
Lucia Gastaldi Università di Brescia 10/30/2010 - 11/7/2010
Joscha Gedicke Humboldt-Universität 10/30/2010 - 12/4/2010
Marc Iwan Gerritsma Technische Universiteit te Delft 10/30/2010 - 11/6/2010
Andrew Kruse Gillette University of Texas at Austin 10/31/2010 - 11/5/2010
Matthias K. Gobbert University of Maryland Baltimore County 11/28/2010 - 12/2/2010
Jay Gopalakrishnan University of Florida 9/1/2010 - 6/30/2011
Shiyuan Gu Louisiana State University 9/1/2010 - 6/30/2011
Thirupathi Gudi Indian Institute of Science 11/27/2010 - 12/3/2010
Johnny Guzman Brown University 11/1/2010 - 11/13/2010
Xiaoming He Missouri University of Science and Technology 11/3/2010 - 11/7/2010
Xiaoming He Missouri University of Science and Technology 11/28/2010 - 12/3/2010
Ying He Purdue University 11/29/2010 - 12/3/2010
Yuan He Columbia University 11/28/2010 - 12/4/2010
Jan S. Hesthaven Brown University 10/30/2010 - 11/6/2010
Robert L. Higdon Oregon State University 10/31/2010 - 11/5/2010
Ronald H.W. Hoppe University of Houston 9/6/2010 - 12/20/2010
Raya Horesh University of Minnesota 10/15/2010 - 11/6/2010
Thomas Yizhao Hou California Institute of Technology 10/30/2010 - 11/4/2010
Jason Howell Clarkson University 10/31/2010 - 11/7/2010
Yulia Hristova University of Minnesota 9/1/2010 - 8/31/2011
Lili Hu Georgia Institute of Technology 10/31/2010 - 11/5/2010
Jae Woo Jeong Miami University 11/5/2010 - 11/6/2010
Sunnie Joshi Texas A & M University 10/30/2010 - 11/5/2010
Lili Ju University of South Carolina 10/31/2010 - 11/4/2010
Myungjoo Kang Seoul National University 10/31/2010 - 11/5/2010
Guido Kanschat Texas A & M University 9/6/2010 - 12/20/2010
Chiu-Yen Kao Ohio State University 9/1/2010 - 12/20/2010
Ohannes Karakashian University of Tennessee 11/4/2010 - 11/6/2010
Markus Keel University of Minnesota 7/21/2008 - 6/30/2011
David Keyes King Abdullah University of Science & Technology 11/28/2010 - 12/3/2010
Chisup Kim Catholic University of America 11/5/2010 - 11/6/2010
Hyunju Kim University of North Carolina - Charlotte 11/4/2010 - 11/6/2010
Seungil Kim Southern Methodist University 11/4/2010 - 11/7/2010
Jung-Han Kimn South Dakota State University 11/29/2010 - 12/2/2010
Axel Klawonn Universität Duisburg-Essen 11/27/2010 - 12/3/2010
Jon Kleinberg Cornell University 11/7/2010 - 11/10/2010
Tzanio V Kolev Lawrence Livermore National Laboratory 11/29/2010 - 12/3/2010
Pawel Konieczny University of Minnesota 9/1/2009 - 8/31/2011
Ralf Kornhuber Freie Universität Berlin 11/28/2010 - 12/5/2010
Kristina Kraakmo University of Central Florida 10/30/2010 - 11/3/2010
Johannes Karl Kraus Johann Radon Institute for Computational and Applied Mathematics 11/27/2010 - 12/3/2010
Christian Kreuzer Universität Duisburg-Essen 11/28/2010 - 12/4/2010
JaEun Ku Oklahoma State University 11/4/2010 - 11/7/2010
Angela Kunoth Universität Paderborn 10/31/2010 - 11/7/2010
Stig Larsson Chalmers University of Technology 11/3/2010 - 11/6/2010
Ilya Lashuk Lawrence Livermore National Laboratory 10/29/2010 - 11/5/2010
Anita Layton Duke University 11/1/2010 - 11/4/2010
Raytcho Lazarov Texas A & M University 11/3/2010 - 11/7/2010
Sabine Le Borne Tennessee Technological University 11/28/2010 - 12/3/2010
Jungho Lee Argonne National Laboratory 11/29/2010 - 12/3/2010
Young-Ju Lee Rutgers University 11/4/2010 - 11/7/2010
Gilad Lerman University of Minnesota 9/1/2010 - 6/30/2011
Dmitriy Leykekhman University of Connecticut 10/31/2010 - 11/7/2010
Dmitriy Leykekhman University of Connecticut 11/29/2010 - 12/3/2010
Fengyan Li Rensselaer Polytechnic Institute 9/1/2010 - 12/20/2010
Hengguang Li University of Minnesota 8/16/2010 - 8/15/2011
Jichun Li University of Nevada 11/2/2010 - 11/6/2010
Jing Li Kent State University 11/28/2010 - 12/4/2010
Lizao (Larry) Li University of Minnesota 11/28/2010 - 12/3/2010
Yan Li University of Minnesota 10/30/2010 - 11/6/2010
Zhilin Li North Carolina State University 10/31/2010 - 11/5/2010
Hyeona Lim Mississippi State University 10/30/2010 - 11/4/2010
Guang Lin Pacific Northwest National Laboratory 10/29/2010 - 11/3/2010
Runchang Lin Texas A&M International University (TAMIU) 11/4/2010 - 11/6/2010
Tao Lin Virginia Polytechnic Institute and State University 11/5/2010 - 11/7/2010
Zhi (George) Lin University of Minnesota 9/1/2009 - 8/31/2011
Hailiang Liu Iowa State University 10/31/2010 - 11/5/2010
Jiangguo (James) Liu Colorado State University 10/31/2010 - 11/7/2010
Xinfeng Liu University of South Carolina 11/1/2010 - 11/4/2010
Irene Livshits Ball State University 11/28/2010 - 12/3/2010
Alexei Lozinski Université de Toulouse III (Paul Sabatier) 10/31/2010 - 11/5/2010
Mitchell Luskin University of Minnesota 9/1/2010 - 6/30/2011
Lina Ma Purdue University 10/31/2010 - 11/6/2010
Scott MacLachlan Tufts University 10/24/2010 - 12/3/2010
Yvon Jean Maday Université de Paris VI (Pierre et Marie Curie) 11/1/2010 - 11/4/2010
Niall Madden National University of Ireland, Galway 10/18/2010 - 12/10/2010
Kara Lee Maki University of Minnesota 9/1/2009 - 8/31/2011
Jan Mandel University of Colorado 11/29/2010 - 12/3/2010
Thomas A. Manteuffel University of Colorado 11/30/2010 - 12/3/2010
Yu (David) Mao University of Minnesota 8/31/2010 - 8/30/2011
Maider Judith Marin-McGee University of Puerto Rico 10/29/2010 - 11/6/2010
Tarek P Mathew University of Colorado 11/28/2010 - 12/4/2010
Jens Markus Melenk Technische Universität Wien 10/30/2010 - 11/7/2010
Irina Mitrea University of Minnesota 8/16/2010 - 6/14/2011
Dimitrios Mitsotakis University of Minnesota 10/27/2010 - 8/31/2011
Peter Monk University of Delaware 9/8/2010 - 12/10/2010
Brian Edward Moore University of Central Florida 10/31/2010 - 11/3/2010
Zhe Nan Louisiana State University 10/30/2010 - 11/7/2010
Michael Joseph Neilan Louisiana State University 10/29/2010 - 11/7/2010
Michael Joseph Neilan Louisiana State University 11/27/2010 - 12/4/2010
Ngoc-Cuong Nguyen Massachusetts Institute of Technology 10/31/2010 - 11/5/2010
Nilima Nigam Simon Fraser University 11/1/2010 - 11/6/2010
Sylvain Nintcheu Fata Oak Ridge National Laboratory 11/1/2010 - 1/29/2011
Victor Nistor Pennsylvania State University 11/4/2010 - 11/7/2010
Ricardo H. Nochetto University of Maryland 9/13/2010 - 12/15/2010
Ioannis Nompelis University of Minnesota 11/28/2010 - 12/3/2010
Hae-Soo Oh University of North Carolina - Charlotte 11/4/2010 - 11/7/2010
Minah Oh James Madison University 11/4/2010 - 11/6/2010
Minah Oh James Madison University 11/28/2010 - 12/4/2010
Luke Olson University of Illinois at Urbana-Champaign 11/28/2010 - 12/5/2010
Alexandra Ortan University of Minnesota 9/16/2010 - 6/15/2011
Cecilia Ortiz-Duenas University of Minnesota 9/1/2009 - 8/31/2011
Miao-Jung Yvonne Ou Oak Ridge National Laboratory 8/30/2010 - 12/10/2010
Jeffrey Ovall University of Kentucky 10/31/2010 - 11/7/2010
Sevtap Ozisik Rice University 11/2/2010 - 11/7/2010
Eun-Hee Park Louisiana State University 10/30/2010 - 11/7/2010
Eun-Hee Park Louisiana State University 11/27/2010 - 12/4/2010
Joseph E. Pasciak Texas A & M University 11/5/2010 - 11/6/2010
Jaime Peraire Massachusetts Institute of Technology 10/31/2010 - 11/5/2010
Ilaria Perugia Università di Pavia 10/30/2010 - 11/6/2010
Petr Plechac University of Tennessee 9/1/2010 - 12/10/2010
Jingmei Qiu Colorado School of Mines 10/31/2010 - 11/3/2010
Weifeng (Frederick) Qiu University of Minnesota 8/31/2010 - 8/30/2011
Vincent Quenneville-Belair University of Minnesota 9/16/2010 - 6/15/2011
Rachel Quinlan National University of Ireland, Galway 10/18/2010 - 12/10/2010
Naveen Ramunigari University of Texas 11/27/2010 - 11/30/2010
Darsh Priya Ranjan University of California, Berkeley 10/29/2010 - 11/6/2010
S. S. Ravindran University of Alabama 11/4/2010 - 11/7/2010
Armin Reiser Louisiana State University 9/1/2010 - 12/13/2010
Fernando Reitich University of Minnesota 9/1/2010 - 6/30/2011
Gianluigi Rozza École Polytechnique Fédérale de Lausanne (EPFL) 10/30/2010 - 11/6/2010
Ulrich Rüde Friedrich-Alexander-Universität Erlangen-Nürnberg 11/28/2010 - 12/3/2010
Giancarlo Sangalli Università di Pavia 10/30/2010 - 11/6/2010
Fadil Santosa University of Minnesota 7/1/2008 - 6/30/2011
Marcus Sarkis Worcester Polytechnic Institute 11/29/2010 - 12/3/2010
Francisco-Javier Sayas University of Delaware 10/28/2010 - 11/7/2010
Alfred Schatz Cornell University 11/4/2010 - 11/7/2010
Reinhold Schneider TU Berlin 10/30/2010 - 11/6/2010
Joachim Schöberl Technische Universität Wien 11/1/2010 - 11/7/2010
Dominik M. Schoetzau University of British Columbia 10/30/2010 - 11/7/2010
Christoph Schwab ETH Zürich 10/31/2010 - 11/7/2010
Marc Alexander Schweitzer Rheinische Friedrich-Wilhelms-Universität Bonn 10/31/2010 - 11/7/2010
Guglielmo Scovazzi Sandia National Laboratories 10/29/2010 - 11/5/2010
Francisco G Serpa Booz Allen Hamilton Inc. (BAH) 11/27/2010 - 12/4/2010
Shuanglin Shao University of Minnesota 9/1/2009 - 8/31/2011
Natasha Shilla Sharma University of Houston 11/5/2010 - 11/7/2010
Mikhail Shashkov Los Alamos National Laboratory 10/31/2010 - 11/5/2010
Jie Shen Purdue University 10/30/2010 - 11/6/2010
Jie Shen Purdue University 11/30/2010 - 12/3/2010
Ke Shi University of Minnesota 10/30/2010 - 11/6/2010
Chi-Wang Shu Brown University 10/31/2010 - 11/6/2010
Ari Stern University of California, San Diego 10/31/2010 - 11/6/2010
Rob Stevenson Universiteit van Amsterdam 10/30/2010 - 11/6/2010
Panagiotis Stinis University of Minnesota 9/1/2010 - 6/30/2011
Jiguang Sun Delaware State University 10/31/2010 - 11/7/2010
Pengtao Sun University of Nevada 11/2/2010 - 11/6/2010
Tong Sun Bowling Green State University 10/30/2010 - 11/7/2010
Yi Sun Statistical and Applied Mathematical Sciences Institute (SAMSI) 10/31/2010 - 11/4/2010
Li-yeng Sung Louisiana State University 9/1/2010 - 6/15/2011
Nicolae Tarfulea Purdue University, Calumet 9/1/2010 - 6/15/2011
Radek Tezaur Stanford University 11/29/2010 - 12/3/2010
Vidar Thomée Chalmers University of Technology 11/3/2010 - 11/7/2010
Dimitar Trenev University of Minnesota 9/1/2009 - 8/31/2011
Ray S. Tuminaro Sandia National Laboratories 11/29/2010 - 12/3/2010
Catalin Turc Case Western Reserve University 10/31/2010 - 11/5/2010
Danail Vassilev University of Pittsburgh 11/4/2010 - 11/6/2010
Panayot S Vassilevski Lawrence Livermore National Laboratory 10/31/2010 - 11/5/2010
Panayot S Vassilevski Lawrence Livermore National Laboratory 11/6/2010 - 11/7/2010
Andreas Michael Veeser Università di Milano 11/27/2010 - 12/4/2010
Chad N Vidden Iowa State University 10/31/2010 - 11/5/2010
Peter Edward Vincent Stanford University 10/31/2010 - 11/6/2010
Michael Vogelius Rutgers University 11/4/2010 - 11/7/2010
Lars B. Wahlbin Cornell University 11/4/2010 - 11/6/2010
Shawn W. Walker Louisiana State University 10/30/2010 - 11/6/2010
Noel J. Walkington Carnegie Mellon University 11/4/2010 - 11/7/2010
Junping Wang National Science Foundation 11/4/2010 - 11/7/2010
Kening Wang University of North Florida 11/29/2010 - 12/3/2010
Wei Wang Florida International University 10/31/2010 - 11/7/2010
Yanqiu Wang Oklahoma State University 11/4/2010 - 11/7/2010
Ying Wang University of Minnesota 11/1/2010 - 11/6/2010
Timothy C. Warburton Rice University 11/1/2010 - 11/4/2010
Yaoguang Wei University of Minnesota 11/28/2010 - 11/29/2010
Olof B. Widlund New York University 11/28/2010 - 12/3/2010
Ragnar Winther University of Oslo 10/17/2010 - 11/12/2010
Barbara Wohlmuth Technical University of Munich 10/30/2010 - 11/6/2010
Carol S. Woodward Lawrence Livermore National Laboratory 11/29/2010 - 12/3/2010
Dexuan Xie University of Wisconsin 11/28/2010 - 12/3/2010
Yulong Xing University of Tennessee 11/1/2010 - 11/7/2010
Jinchao Xu Pennsylvania State University 11/5/2010 - 11/7/2010
Jinchao Xu Pennsylvania State University 11/29/2010 - 12/2/2010
Liwei Xu Rensselaer Polytechnic Institute 10/31/2010 - 11/7/2010
Guangri Xue University of Texas at Austin 10/29/2010 - 11/7/2010
Sergey Borisovich Yakovlev Rensselaer Polytechnic Institute 9/8/2010 - 12/15/2010
Jue Yan Iowa State University 10/31/2010 - 11/7/2010
Ulrike Meier Yang Lawrence Livermore National Laboratory 11/28/2010 - 12/3/2010
Xingzhou Yang Mississippi State University 10/29/2010 - 11/4/2010
Irad Yavneh Technion-Israel Institute of Technology 11/28/2010 - 12/4/2010
Xiu Ye University of Arkansas 10/31/2010 - 11/7/2010
Haijun Yu Purdue University 11/29/2010 - 12/3/2010
Hui Yu Iowa State University 10/31/2010 - 11/5/2010
Suxing Zeng Wright State University 11/28/2010 - 12/3/2010
Shangyou Zhang University of Delaware 11/5/2010 - 11/6/2010
Shangyou Zhang University of Delaware 11/28/2010 - 12/4/2010
Yi Zhang Louisiana State University 11/27/2010 - 12/4/2010
Zhimin Zhang Wayne State University 10/31/2010 - 11/7/2010
Shan Zhao University of Alabama 10/31/2010 - 11/2/2010
Huiqing Zhu University of Southern Mississippi 11/5/2010 - 11/7/2010
Ludmil Zikatanov Pennsylvania State University 11/27/2010 - 12/3/2010
Legend: Postdoc or Industrial Postdoc Long-term Visitor

IMA Affiliates:
Arizona State University, Boeing, Corning Incorporated, ExxonMobil, Ford, General Motors, Georgia Institute of Technology, Honeywell, IBM, Indiana University, Iowa State University, Korea Advanced Institute of Science and Technology (KAIST), Lawrence Livermore National Laboratory, Lockheed Martin, Los Alamos National Laboratory, Medtronic, Michigan State University, Michigan Technological University, Mississippi State University, Northern Illinois University, Ohio State University, Pennsylvania State University, Portland State University, Purdue University, Rice University, Rutgers University, Sandia National Laboratories, Schlumberger Cambridge Research, Schlumberger-Doll, Seoul National University, Siemens, Telcordia, Texas A & M University, University of Central Florida, University of Chicago, University of Delaware, University of Houston, University of Illinois at Urbana-Champaign, University of Iowa, University of Kentucky, University of Maryland, University of Michigan, University of Minnesota, University of Notre Dame, University of Pennsylvania, University of Pittsburgh, University of Tennessee, University of Wisconsin, University of Wyoming, US Air Force Research Laboratory, Wayne State University, Worcester Polytechnic Institute