# <span class=strong>Reception and Poster Session</span><br><br/><br/><b>Poster submissions welcome from all participants</b><br><br/><br/><a<br/><br/>href=/visitor-folder/contents/workshop.html#poster><b>Instructions</b></a><br/><br/>

Monday, November 1, 2010 - 5:00pm - 7:00pm

Lind 409

**Application of DPG method to hyperbolic problems**

Leszek Demkowicz (The University of Texas at Austin)

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.**A C0 interior penalty method for a biharmonic problem**

with essential and natural boundary conditions of Cahn-Hilliard

type

Shiyuan Gu (Louisiana State University)

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 C^{0}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.**HDG methods for CFD applications**

Ngoc-Cuong Nguyen (Massachusetts Institute of Technology)

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.**A multipoint flux mixed finite element method on general**

hexahedra: Multiscale mortar extension and applications to

multiphase flow in porous media

Guangri Xue (The University of Texas at Austin)

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.**Isogeometric Analysis for electromagnetic problems**

Giancarlo Sangalli (Università di Pavia)

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.**Cochain interpolation for spectral element methods**

Marc Gerritsma (Technische Universiteit te Delft)

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.**Application of DPG method to wave propagation**

Leszek Demkowicz (The University of Texas at Austin)

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.**Solution of dual-mixed elasticity equations**

using Arnold-Falk-Winther element and

discontinuous Petrov-Galerkin method.

A comparison

Leszek Demkowicz (The University of Texas at Austin)

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.**Numerical simulation of three-dimensional nonlinear water waves**

Liwei Xu (Rensselaer Polytechnic Institute)

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.**HDG methods for multiphysics simulation**

Ngoc-Cuong Nguyen (Massachusetts Institute of Technology)

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 thek+1 for diffusion-dominated problems; (3) in

optimal order

such cases, local postprocessing can be developed toto k+2 for the approximation

increase the convergence rate

of the field variables; (4) they can deal with; (5) they result in

non-compatible boundary conditions

aand (6) they are somewhat easier to

compact matrix system

implement and provide a single code for solving multiphysics

problems.**Spectral methods for systems of coupled equations and**

applications to Cahn-Hilliard equations

Feng Chen (Purdue University)

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.**Finite element methods for the Monge-Ampere equation**

Michael Neilan (Louisiana State University)

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.**GPU accelerated discontinuous Galerkin methods**

Timothy Warburton (Rice University)

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.**Multiscale methods for complex systems**

Angela Kunoth (Universität Paderborn)

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.**Numerical smoothness and error analysis for RKDG**

on the scalar nonlinear conservation laws

Tong Sun (Bowling Green State University)

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.**Pseudo-time continuation and time marching methods for Monge-Ampère type equations**

Gerard Awanou (Northern Illinois University)

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.**The flux reconstruction approach to high-order methods:**

Theory

and application

Peter Vincent (Stanford University)

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.**Adaptive solution of parametric eigenvalue problems for partial differential equations**

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.**Conformal conservation laws and geometric integration**

for Hamiltonian PDE with added dissipation

Brian Moore (University of Central Florida)

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.**Properties of the volume corrected characteristic mixed method**

Todd Arbogast (The University of Texas at Austin)

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.**Local properties of finite element solutions for advection-dominated optimal control problems**

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.**The generalized fundamental theorem of calculus and its applications to boundary element methods**

Sylvain Nintcheu Fata (Oak Ridge National Laboratory)

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.**Error estimates for generalized barycentric interpolation**

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.**Shape optimization of chiral propellers in 3-D stokes**

flow

Shawn Walker (Louisiana State University)

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.**Numerical study of singular solutions of relativistic Euler equations**

Yi Sun (Statistical and Applied Mathematical Sciences Institute (SAMSI))

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.**Space-time sparse wavelet FEM for parabolic equations**

Roman Andreev (ETH Zürich)

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.**Efficient, accurate and**

rapidly-convergent algorithms for the solution of three dimensional acoustic and electromagnetic scattering problems in domains with geometric singularities

Catalin Turc (Case Western Reserve University)

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).**Central discontinuous Galerkin methods for ideal MHD equations**

with the exactly divergence-free magnetic field

Liwei Xu (Rensselaer Polytechnic Institute)

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.**Reduced-order modelling for electromagnetics**

Yanlai Chen (University of Massachusetts Dartmouth)

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.**Computational aspects of a two-scale finite element**

method for singularly perturbed problems

Niall Madden (National University of Ireland, Galway)

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(N^{2}) to O(N^{3/2}).

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.**Positivity-preserving discontinuous Galerkin schemes**

for linear Vlasov-Boltzmann transport equations

Yingda Cheng (Brown University)

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.**High order well-balanced schemes for non-equilibrium flows**

Wei Wang (Florida International University)

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.**Hodge decomposition and Maxwell's equations**

Jintao Cui (University of Minnesota, Twin Cities)

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.**Point-wise hierarchical reconstruction for discontinuous**

Galerkin and finite volume methods for solving conservation

laws

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.**High order integral deferred correction method based on Strang split semi-Lagrangian WENO method for Vlasov Poisson simulations**

Jingmei Qiu (Colorado School of Mines)

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.**Wavenumber-explicit convergence analysis for the**

Helmholtz equation: hp-FEM and hp-BEM

Jens Melenk (Technische Universität Wien)

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).**Multiscale finite element method in perforated domains**

Alexei Lozinski (Université de Toulouse III (Paul Sabatier))

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.**A reduced basis hybrid method for viscous flows in**

parametrized complex networks

Gianluigi Rozza (École Polytechnique Fédérale de Lausanne (EPFL))

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.**Application of DPG method to Stokes equations**

Leszek Demkowicz (The University of Texas at Austin)

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.