<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
- 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,
systems of hyperbolic equations and the compressible Euler
- A C0 interior penalty method for a biharmonic problem
with essential and natural boundary conditions of Cahn-Hilliard
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 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.
- 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
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
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.
 Marc Gerritsma, Edge functions for spectral element methods,
Proceedings of ICOSAHOM 2009, 2010.
 Mick Bouman, Artur Palha, Jasper Kreeft and Marc Gerritsma, A
conservative spectral element method for curvilinear domains,
Proceedings of ICOSAHOM2009, 2010.
 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.
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
- 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 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
compact matrix system and (6) they are somewhat easier to
implement and provide a single code for solving multiphysics
- 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
- 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:
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
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
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 , 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.
 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
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
wave numbers k. In order to understand how the wave number
the convergence properties of discretizations of such problems,
we develop a regularity theory for the Helmholtz equation that
in k. At the heart of our analysis is the decomposition of
into two components: the first component is an analytic, but
highly oscillatory function and the second one has finite
features wavenumber-independent bounds.
This understanding of the solution structure opens the door to
analysis of discretizations of the Helmholtz equation that are
in their dependence on the wavenumber k.
As a first example, we show for a conforming high order finite
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
kh/p is small.
As a second example, we consider combined field boundary
arising in acoustic scattering. Also for this example, the same
conditions as in the high order finite element case suffice to
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,
- 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