# Poster Session and Reception

Tuesday, May 11, 2004 - 3:40pm - 6:00pm

Lind 400

**Numerical Homogenization of Nonlinear Partial Differential Equations and its Applications**

Yalchin Efendiev (Texas A & M University)

The numerical homogenization methods presented in this talk are designed to compute homogenized solutions. In particular we are interested when the heterogeneities have random nature. I will describe numerical homogenization methods that we proposed recently and their relation to some other multiscale methods. Convergence of these methods for nonlinear parabolic equations will be discussed. Numerical examples and applications will be considered.**New Mimetic Discretizations of Diffusion-Type Problems on Polygonal Meshes**

Konstantin Lipnikov (Los Alamos National Laboratory)

Joint work with Yuri Kuznetsov and Mikhail Shashkov.

The determining factor for reliability, accuracy, and efficiency of simulations is accurate locally conservative discretizations. Practice experience shows that the most effective discrete approximations preserve and mimic the underlying properties of original continuum differential operators. One of such approaches, the mimetic finite difference technique based on the support-operator methodology, has been applied successfully to several applications including diffusion, electromagnetics and gas dynamics.

As mathematical modeling becomes more sophisticated, the need for discretization methods handling meshes with mixed types of elements has arisen. On this poster we present new mimetic discretizations on polygonal meshes. AMR meshes, non-matching meshes and meshes with non-convex cells are important examples of polygonal meshes.

Nowadays, a limited use of polygonal meshes is restricted by a small number of accurate discretization schemes. We describe the new mimetic discretizations for a diffusion-reaction problem formulated as a system of two first-order equations. The discretization technique results in a method which is exact for linear solutions. The method is second order accurate for general problems with or without material discontinuities and relatively easy to solve (it produces a symmetric positive definite matrix). The new discretization technique can be extended to polyhedral meshes and some other PDEs.**Towards a Variational Complex of the Finite Element Method**

Elizabeth Mansfield (University of Kent at Canterbury)

Exact differential complexes are important in the design of finite element approximate schemes. This poster starts with these and discusses how they may be extended to a full variational complex. The motivation is to be able to answer the question, Can you design a finite element scheme for a system which inherits both variational principle and certain pre-selected conservation laws exactly? This is joint work with Reinout Quispel (Latrobe University, Australia).**Compatible Discretizations in Lagrangian/Eulerian Resistive MHD Modeling for Z-pinch Applications**

Allen Robinson (Sandia National Laboratories)

We give an overview of the use of compatible discretization techniques used in resistive magnetohydrodynamic (MHD) modeling for Z-pinch simulations at Sandia National Laboratories. Z-pinch MHD physics is dominated by moving material regions whose conductivity properties vary drastically as material passes through melt and plasma regimes. At the same time void regions are modeled as regions of very low conductivity. This challenging physical situation requires a sophisticated modeling approach matched by sufficient computational resources to make progress in physical understanding. An Arbitrary-Lagrangian-Eulerian (ALE) operator split methodology for modeling the MHD equations on unstructured grids is described. An implicit treatment of the magnetic diffusion equation, represented using low order vector edge and face elements, gives solutions free from parasitic transients. A matching algebraic multigrid must also be applied to deal with the large null space of the stiffness matrix. We also discuss the important isssue of constrained transport remapping on unstructured grids and how this can be viewed in the framework of low order face elements. An important issue to be resolved associated with the constrained transport algorithm is the accurate reconstruction of the magnetic flux density field given information on the fluxes on element faces.

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.**Eigenvalue Solvers for Electromagnetic Fields in Cavities**

Peter Arbenz (Zentrum für Umfragen, Methoden und Analysen (ZUMA) e.V.)

Joint work with M. Becka, Institute of Computational Science, ETH Zurich R. Geus, Paul-Scherrer Institute, Villingen U. Hetmaniuk, Sandia National Laboratories, Albuquerque.

We investigate the Jacobi-Davidson algorithm for computing a few of the smallest eigenvalues of a generalized eigenvalue problem resulting from the finite element discretization of the time-harmonic Maxwell equation. Various multilevel preconditioners are employed to improve the convergence rate and memory consumption of the eigensolver. We present sequential results of very large eigenvalue problems originating from the design of resonant cavities of particle accelerators. Furthermore we detail our approach for parallelizing our code by means of the Trilinos software framework.**A Vertex-Centered Dual Discontinuous Galerkin Method**

Martin Berggren (Uppsala University)

I will present a new discontinuous Galerkin method for discretizing partial differential equations with a dominating hyperbolic character. At lowest order, the method reduces to a vertex-centered finite-volume method with control volumes based on a dual mesh, and the method can be implemented using an edge-based data structure. Preliminary tests on a model linear hyperbolic equation in 2D indicate a favorable qualitative behavior for nonsmooth solutions, and an optimal convergence rate for smooth solutions when using locally piecewise-linear approximations.**A Finite Volume Element Method for a Nonlinear Elliptic Problem**

Panagiotis Chatzipantelidis (Texas A & M University)

Joint work with: V. Ginting and R.D. Lazarov.

We consider a finite volume discretization of second order nonlinear elliptic boundary value problems on polygonal domains. For sufficiently small data, we show existence and uniqueness of the finite volume solution using a fixed point iteration method. We derive error estimates in H1-, L2 and Linfinity-norm. In addition a Newton's method is analyzed for the approximation of the finite volume solution and numerical experiments are presented.**De Rham Diagram for Projection-Based Interpolation. 3D Optimal p- and hp-Error Estimates**

Leszek Demkowicz (The University of Texas at Austin)

I will present the main idea and results for the commuting de Rham diagram for polynomial spaces corresponding to 3D finite elements of variable order generalizing Nedelec tetrahedrons of the first and second type, hexahedron of the first type (Nedelec's hexahedron of the second type does not satisfy the commuting diagram property), and prisms of the first and second type.

I will discuss shortly the generalization to parametric elements, including the most popular isoparametric elements.

The optimal p-interpolation (and following hp-interpolation) estimates are obtained by comparing the interpolation errors with (commuting) projections errors. Two fundamental tools necessary for the interpolation error analysis include recent results on existence of continuous, polynomial preserving extension operators (with M. Ainsworth) and discrete Friedrichs inequalities (with J. Gopalakrishnan). The presented interpolation theory summarizes a recent work done with A. Buffa. The methodology differs considerably from earlier 2D results obtained with I. Babuska.**Mixed Discretisation Methods for Discontinuous Galerkin Method with Analytical Test-Functions**

Juergen Geiser (Texas A & M University)

Joint work with: R. Lazarov and R. Ewing.

Our mathematical models describe transport and reaction processes in porous media. Based on our model equations we present a new mixed discretization methods with analytical test-functions and the error-analysis. The based convection-diffusion-reaction-equations are discretised with Discontinuous Galerkin methods in a mixed formulation (LDG-method).

We introduce the variational formulations and our adjoint problem to derive the analytical solutions for the test-functions. The stability of the discretization methods are discussed and an abstract error-estimates is derived. We apply the abstract error-estimates for the different test-functions, e.g. polynomial- and exponential-functions and present an improved optimal order result for our new exponential test-functions.

The application for our new discretization methods are proposed. Finally we discuss our further works.

Key words: convection-diffusion-dispersion-reaction-equation, Discontinuous Galerkin method, analytical methods**Discrete Connections on Principal Bundles**

Melvin Leok (California Institute of Technology)

Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding to the choice of a splitting of the short exact sequence.

Equivalent representations of a discrete connection are considered, and an extension of the pair groupoid composition, that takes into account the principal bundle structure, is introduced. Computational issues, such as the order of approximation, are also addressed. Discrete connections provide an intrinsic method for introducing coordinates on the reduced space for discrete mechanics, and provide the necessary discrete geometry to introduce more general discrete symmetry reduction.

In addition, discrete analogues of the Levi-Civita connection, and its curvature, are introduced by using the machinery of discrete exterior calculus, and discrete connections.

This is part of a broader program to develop discrete analogues of differential geometry which are relevant to the systematic construction of computational geometric mechanics.

Joint work with Jerrold E. Marsden (Caltech) and Alan D. Weinstein (Berkeley).**Discrete Exterior Calculus and its Applications in Mechanics and Computer Science**

Anil Hirani (California Institute of Technology)

To solve PDEs on irregular, non-flat meshes, one can either interpolate and compute differential quantities, or define a discrete calculus without interpolation while preserving some of the structure of the smooth theory. Discrete exterior calculus (DEC) is tensor analysis on irregular, simplicial meshes and their duals, which takes the latter approach. It defines differential operators on such meshes in a coordinate independent way. With it, one can solve, for example, Laplace's equation on an arbitrary triangle mesh surface. Most numerical methods except FEM, are for flat, regular grids. While FEM involves interpolation of scalar values on irregular grids, DEC works with interpolations of values defined on points, edges, triangles etc. In addition, it provides a way to incorporate vector fields into such a framework, thus allowing for computations on moving meshes. We will describe DEC and suggest some of its applications in computer science and mechanics. For more information, see http://www.cs.caltech.edu/~hirani**Intuitive vs. Computable Topological Aspects of Computational Electromagnetics**

P. Robert Kotiuga (Boston University)

Intuitive problems, such as checking if a space is contractible, are easily characterized in terms of homotopy groups but, in four or more dimensions, such a characterization is provably computationally intractable. On the other hand, cohomology theory may not be intuitive, but it does provide a formal connection between Maxwell's equations and the lumped parameters occurring in Kirchhoff's laws. Also, cohomological information is efficiently extracted from the data structures used in finite element analysis. A natural question is: Do engineers need to go beyond the linear algebra and sparse matrix techniques associated with homology calculations? It turns out that there are inverse problems involving near force-free magnetic fields where the conjectured characterization of the space of solutions, involves computationally intractable topological invariants. Hence, it is imperative to investigate algebraic structures found in the data structures of finite element analysis, which yield topological insights not deducible from cohomological considerations alone.

The Hurewicz map is a well-defined map taking representatives of generators of homotopy groups to their homology classes. In this sense, it provides a natural framework for comparing the intuitive but intractable with the computable but less intuitive. The presentation will develop this theme in the context of computational electromagnetics.

Recent work:

1) Kotiuga, P. R., Topology-Based Inequalities and Inverse Problems for Near Force-Free Magnetic Fields, IEEE Trans. MAG. March 2004.

2) Gross, P.W., Kotiuga, P.R., Electromagnetic Theory and Computation: A Topological Approach. MSRI Monograph series # 48; Cambridge U. P., 2004. ISBN # 0521801605.

3) Suuriniemi, S., Kettunen, L., Kotiuga, P.R., Techniques for Systematic Treatment of Certain Coupled Problems. IEEE Trans. MAG-39, (3), May 2003, pp 1737-1740.**Why Mixed Finite Elements are not used in the Petroleum Industry and what can we do about it?**

Ilya Mishev (ExxonMobil)

The purpose of this poster is to invigorate the dialogue between the academia and the industry. We start with a short description of the most common formulation used in the petroleum industry to model the fluid flow in porous media and discuss what are the implications for the discretizations. One approach based on primal dual Mixed Finite element method will be considered and some examples given.**Mimetic Preconditioners for Mixed Discretizations of the Diffusion Equation**

J. David Moulton (Los Alamos National Laboratory)

Joint work with Travis M. Austin, M. Shashkov, and Jim E. Morel.

Mixed discretizations (e.g., mimetic, or compatible) are based on the first order form, and hence, naturally lead to an indefinite linear system. Although optimal preconditioners have been developed for the case of orthogonal grids and a diagonal diffusion tensor, the performance of these methods degrades with full tensor anisotropy or severe grid distortion. Thus, a significant hurdle in the widespread adoption of these discretization methods is the lack of robust and efficient solvers for the corresponding linear system. To this end we are motivated by one specific advantage that the hybrid or local forms of mixed discretizations exhibit, namely, their more localized sparsity structure. Specifically, for the support operator method (SOM) we consider augmentation of the flux (i.e., vector unknowns) such that an appropriate ordering of the augmented flux leads to a new block diagonal system for this component. In contrast to the block diagonal structure of the hybrid system this system has blocks centered about vertices, and block elimination of the flux (i.e., formation of the Schur complement) leads to a symmetric positive definite scalar problem with a standard cell-based 9-point structure (in two dimensions). This reduced system is readily solved with existing robust multigrid methods, such as Dendy's Black Box Multigrid (BoxMG). An analogous approach is used to augment the hybrid or local SOM system and derive the equivalent preconditioner for this case. We demonstrate the effectiveness of this preconditioner for logically rectangular severely distorted grids.**High Order Symplectic Integration Methods for Finite Element Solutions to Time Dependent Maxwell Equations**

Robert Rieben (Lawrence Livermore National Laboratory)

We motivate the use of high order integration methods for compatible finite element solutions of the time dependent Maxwell equations. In particular, we present a symplectic algorithm for the integration of the coupled first order Maxwell equations for computing the time dependent electric and magnetic fields in a mixed finite element approach. Symplectic methods have the benefit of conserving total electromagnetic field energy and are therefore preferred over dissipative methods (such as traditional Runge-Kutta) in applications that require high-accuracy and energy conservation over long periods of time integration. We present a conditionally stable, explicit time integration scheme that is up to 5th order accurate along with some numerical examples which demonstrate the superior performance of high order time integration methods.**Two-Phase Flow Modeling**

Beatrice Riviere (University of Pittsburgh)

This poster presents a high order finite element method that naturally handles unstructured meshes and heterogeneous porous media for solving the incompressible two-phase flow problem. In the proposed algorithm, the primary variables are the wetting phase pressure and saturation. They are approximated by discontinuous polynomials of varying degree. The flexibility of discontinuous Galerkin methods has made these methods competitive for modeling flow and transport problems. Some of the advantages include the high order approximation, the easy implementation on unstructured grids, the robustness of the method for equations with discontinuous coefficients and the local mass conservation property. Numerical simulations are given for homogeneous and heterogeneous porous media.**Finite Volume Element Methods for Parabolic Integro-Differential Equation with Nonsmooth Initial Data**

Rajen Sinha (Texas A & M University)

Joint work with R. D. Lazarov and R. E. Ewing.

Mathematical models describing the nonlocal reactive flows in porous media and heat conduction through materials with memory give rise to parabolic integro-differential equation. We present a semidiscrete finite volume element(FVE) approximations to parabolic integro differential equation(PIDE) in a two-dimensional convex polygonal domain. More precisely, for homogeneous equation, an elementary energy technique is used to derive optimal error estimate in L^{2}and H^{1}norms for positive time when the given initial function is in H_{0}^{1}.**Finite Element Methods for Non-elliptic but Coercive Problems**

Jean-Marie Thomas (Université de Pau et des Pays de l'Adour)

One analyze finite element methods for a variational problem : find u in V such that

a(u,v) - k^{2}(u,v) = l(v) for any v in V ,

where (.,.) is the scalar product of an Hilbert space H, V is an Hilbert space with continuous imbedding of V into H and V is dense in H, a(.,.) is a symmetric non-negative continuous bilinear form on V x H such that a(.,.) + (.,.) is V-elliptic. Moreover we assume that k 2 is not an eigenvalue of the associated spectral problem. By the finite element method, the discrete problem associated to a finite dimensional subspace of V consists to find uh in Vh such that

a(uh,vh) - k^{2}(u_{h},v_{h}) = l(v_{h}) for any v_{h}in V_{h},

One look for an optimal a priori error bound similar to the Céa Lemma for elliptic problems. In some situations, this result is obtained as soon as h is sufficiently small. In the other situations, we show what additional property must verify Vh for obtaining again the optimal a priori error bound.

Applications to the analysis of some time-harmonic systems from elastodynamics , aeroacoustics and electromagnetism will be presented. At last, primal and dual finite element approximations of the Helmholtz equation will be considered.**Consistent Material Operators for Geometrical Discretization Methods on Generalized Grids**

Rolf Schuhmann (TU Darmstadt)

Joint work with Marco Cinalli and Thomas Weiland.

Geometrical methods for the spatial discretization of Maxwell's equation are able to preserve important properties, like the conservation of charge and energy and the orthogonality of solution spaces. The Finite Integration Technique (FIT) provides for a natural and efficient notation of such approaches, introducing separate matrix operators for (exact) topological and (approximate) material relations. The implementation of the material operators strongly depends on the type of computational grids, but must generally fulfill important consistency and stability properties. In this paper we investigate the accuracy and efficiency of some recently developed material operators for tetrahedral grids. Additionally it will be shown that the requirement for consistency leads to some surprising consequences for the overall simulation scheme.**Formal Theory of PDEs and simulation of Fluid Flows**

Jukka Tuomela (University of Joensuu)

Joint work with Bijan Mohammadi.

In many physical models there appear constraints or conserved quantities which make the system essentially overdetermined. We propose a ne w approach to solve numerically these kind of systems of PDEs which is based on the formal theory of PDEs. The idea is to find the involut ive form of the system, and use it explicitly in the computations. The involutive form is important because many properties of the system cannot be determined if the system is not involutive. We illustrate our approach by considering a compressible flow problem.**Numerical Simulation of Contraints Preserving Boundary Conditions for Constrainted Hyperbolic Equations**

Jing Wang (University of Minnesota, Twin Cities)

Poster