|
Probability
and Statistics in Complex Systems: Genomics, Networks, and Financial
Engineering, September 1, 2003 - June 30, 2004
Abstracts:
IMA
"Hot Topics" Workshop:
May
11-15, 2004
Photo
Gallery Material
from Talks
Ivar
Aavatsmark
(Senter for integrert petroleumsforskning, Universitetet i Bergen)
ivar.aavatsmark@cipr.uib.no
MPFA Discretization on Quadrilateral Grids
Slides:
pdf
The presentation will give an overview of the MPFA methods and
its applications in multiphase flow in porous media. Recent
progress on the theoretical side will be covered: Convergence
proof for one MPFA method, monotonicity properties in the homogeneous
case, new variants of the MPFA method, convergence properties
when the Edwards splitting is used as preconditioner.
Peter
Arbenz (Institute of Computational Science,
Swiss Federal Institute of Technology (ETH Zurich)) arbenz@inf.ethz.ch
http://www.inf.ethz.ch/~arbenz
Eigenvalue
Solvers for Electromagnetic Fields in Cavities (poster
session)
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.
Douglas
N. Arnold (IMA, University of Minnesota) director@ima.umn.edu
http://www.ima.umn.edu/
Differential Complexes and Stability of Finite Element
Methods
Slides:
pdf
Many of the partial differential equations of mathematical physics
are related to differential complexes which determine their
structure and well-posedness. Many successful finite element
discretizations of these problems can best be understood as
arising from piecewise polynomial subcomplexes. The stability
of these methods is obtainted by relating the discrete subcomplex
to the continuous differential complex via a commuting diagram.
The best known case is the de Rham complex, which underlies
both electromagnetic and diffusion problems. In this case there
are a large number of possible piecewise polynomial subcomplexes
of each order. These can be presented systematically using the
Koszul complex. The elasticity equations are related to another
differential complex which can be related to the de Rham complex
through a subtle homological construction. This has lead to
recent progress in the design of stable mixed finite elements
for elasticity in two and three dimensions.
Timothy
J. Barth (NASA Ames Research Center) barth@nas.nasa.gov
Entropy
Stable FEM Discretization of First-Order Systems of Conservation
Laws (pdf)
Slides: pdf
Martin
Berggren
(Department of Scientific Computing, Uppsala University) Martin.Berggren@it.uu.se
A Vertex-Centered Dual Discontinuous Galerkin Method
(poster session)
Slides:
pdf
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.
Pavel
Bochev (Computational Mathematics and Algorithms,
Org. 9214 and Computer Science Research Institute, Sandia National
Laboratories) pbboche@sandia.gov
Variational and Geometric Aspects of Compatible Discretizations
Slides:
html
pdf
ps
ppt
PDE models arise in virtually all fields of science and engineering.
Their compatible discretizations are finite dimensional models
of the physical process that are stable and provide accurate
and physically meaningful solutions.
Variational principles take advantage of the intrinsic connections
between the structure of many PDEs and optimization problems
to identify their compatible discretizations.
Differential complexes provide another tool to encode the structure
of a PDE. Differential forms represent global quantities rather
than fields, and provide a model for the way we observe the
physical process. The idea that differential forms can and should
be used to develop compatible (mimetic) discretizations started
to permeate computational sciences approximately two decades
ago and led to fundamental advances in computational electromagnetics.
Since then, geometrical approaches to discretization have enjoyed
a steady and ever increasing interest and appreciation in computational
sciences. The goal of this lecture is two-fold. I will show
how variational and geometric techniques can complement each
other in the quest for accurate and stable discretizations by
providing tools for the analysis and the design of compatible
models. In doing so I will retrace the key steps that have led
to our modern understanding of these connections. To illustrate,
as well as to compare and contrast variational identifications
of compatibility, I will use the Kelvin and the Dirichlet principles.
Factorization diagrams will reveal the geometrical structure
of the problem and form the basis for the design of compatible
discretizations. Then, with the help of the grid decomposition
property and the commuting diagram property I will show the
fundamental links between the two approaches.
I will conclude with examples of alternative discrete models
that circumvent the rigid structural constraints imposed by
compatibility and talk about the advantages and the perils associated
with their use.
Daniele
Boffi (Dipartimento di Matematica Università di Pavia)
boffi@dimat.unipv.it
http://www-dimat.unipv.it/boffi/
Compatible Discretizations for Eigenvalue Problems
Slides:
pdf
We start with a short review of standard Galerkin approximation
of variationally posed eigenvalue problems, where we mainly
consider the case of elliptic problems associated with a compact
resolvent. Our basic example is the Laplace/Poisson eigenvalue
problem. The main result, in this case, can be summarized by
the claim that any choice of discrete space sequence, which
provides a convergent scheme for the source problem, automatically
performs well when applied to the corresponding eigenvalue problem.
The main discussion in this talk focuses on the approximation
of eigenvalue problems in mixed form. Using again the basic
example of the Laplace/Poisson eigenvalue problem, we show that,
when using a mixed method for its discretization, the picture
is somewhat different from the previous case. The main (counter)
example is given by a choice of discrete space sequences such
that the classical Brezzi's conditions are satisfied (whence
the source problem is correctly approximated) but, when the
eigenvalue problem is considered, several spurious eigenmodes
pollute the discrete spectrum. This surprising behavior is proved
theoretically and numerically demonstrated. We then review the
theory of the discretization of eigenvalue problems in mixed
form (joint work with F. Brezzi and L. Gastaldi), where the
compatibilities between the discrete space sequences for the
good approximation of eigenpairs are made clear. Necessary and
sufficient conditions are given for several general cases of
interest. If time permits, some consequences of the theory can
be presented. One application of the theory is the approximation
of interior Maxwell's eigenvalue (leading in particular to the
discretization of time-harmonic Maxwell's system). The results
can be suitably modified to include the approximation of band
gaps for photonic crystals (joint work with M. Conforti and
L. Gastaldi). Another important consequence concerns the approximation
to evolution problems in mixed form (joint work with L. Gastaldi).
Alain
Bossavit (Laboratoire de Génie Electrique de Paris
(LGEP)) bossavit@lgep.supelec.fr
Computational Electromagnetism and Whitney Forms
Slides:
IMA1.pdf
IMA2.pdf
This talk will give an overview of the use of Whitney forms
in electromagnetism, from 1980 to now, and of desirable developments.
Edge elements (not to be known by that name until about 1986)
allowed to solve eddy current problems in dimension 3, a notorious
conundrum during the 70's. Within ten years, edge element discretization,
Galerkin style, became the established method in computational
electromagnetism, (CEM) while efforts to answer the question
"why edge elements?" slowly fostered familiarity with Whitney
forms and cohomology in the CEM community. Once the central
place of cohomology and commutative diagrams was acknowledged,
ideas about "mimetic" discretizations, equivalent network methods,
etc., could develop in the 90's, and the order of the day, now,
is to found an appropriate "discrete exterior calculus" -- which
holds promises far beyond electromagnetics. Lots of questions
remain, however: Convergence issues, Whitney forms on non-simplicial
meshes, in particular, will be addressed. (Some new results
about the geometric interpretation of degrees of freedom for
higher-degree Whitney forms will be presented.) As for the future,
the demands of engineering about coupled problems (such as magneto-elasticity,
MHD, etc.) will promote a better understanding of the differential-geometric
structures which underlie both electromagnetism and continuum
mechanics.
Panagiotis
Chatzipantelidis (Department of Mathematics,
Texas A&M University) chatzipa@math.tamu.edu
http://www.math.tamu.edu/~chatzipa
A
Finite Volume Element Method for a Nonlinear Elliptic Problem
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 L -norm.
In addition a Newton's method is analyzed for the approximation
of the finite volume solution and numerical experiments are
presented.
Snorre
H. Christiansen
(Centre of Mathematics for Applications (CMA, Department of
Mathematics, University of Oslo) snorrec@math.uio.no
http://folk.uio.no/snorrec/
Div-curl
Lemma for Edge Elements (poster
session)
Slides:
pdf
ps
A
variant of Murat and Tartar's div-curl lemma is stated and proved
for Nedelec's edge elements. In the Galerkin finite element
setting one can expect control of L2 norms of vectorfields
and also sufficient control of their curls in H-1.
But the divergence is usually just controlled trough the integral
of the vectorfields against a finite dimensional space of gradients.
The proposed lemma is designed to handle this case. The proof
uses a uniform norm equivalence related to discrete compactness
properties of vector FE spaces and a super-approximation property
of scalar FE spaces.
Leszek
Demkowicz
(Texas Institute for Computational Engineering Sciences, The
University of Texas at Austin) leszek@ices.utexas.edu
http://www.ices.utexas.edu/~leszek/
De
Rham Diagram for Projection-Based Interpolation. 3D Optimal
p- and hp-Error Estimates ps
format (poster session)
Slides:
pdf
ps
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.
Yalchin
Efendiev (Department
of Mathematics, Texas A&M University) efendiev@math.tamu.edu
Numerical
Homogenization of Nonlinear Partial Differential Equations and
its Applications
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.
Juergen
Geiser (IWR, University of Heidelberg, Visiting
professor of ISC, Texas A&M University, College Station)
jgeiser@math.tamu.edu
Mixed
Discretisation Methods for Discontinuous Galerkin Method with
Analytical Test-Functions (poster
session)
Paper:
pdf
ps
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
Ralf
Hiptmair
(SAM - Seminar for Applied Mathematics, ETH-Zentrum) hiptmair@sam.math.ethz.ch
http://www.sam.math.ethz.ch/~hiptmair
Higher
Order Whitney Forms (pdf
version)
Slides: pdf
Compatible
finite element discretizations of second-order boundary value
problems set in the function spaces H1( ),
H(curl, ),
and H(div, )
will naturally rely on discrete differential forms. Their lowest-order
representatives are the well-knowns linear Lagrangian finite
elements, edge elements, and face elements. However, given the
interior smoothness of the solutions of many boundary value
problems, approximation by local polynomials of high degree
offers superior efficiency (hp-version of FEM).
The
first part of the presentation will focus on the algebra behind
the construction of higher order Whitney forms. The calculus
of differential forms will be used to devise a unified description
of discrete differential forms of any order and polynomial degree
on simplicial meshes in any spatial dimension. A general formula
for suitable degrees of freedom is also available. Fundamental
properties of nodal interpolation can be established easily.
It turns out that higher order spaces, including variants with
locally varying polynomial order, emerge from the usual Whitney-forms
by local augmentation.
The
second part of the talk will review the concrete construction
of suitable basis functions for higher order Whitney forms.
Discrete 1-forms will serve as main example. Recent inventions
of p-hierarchical bases with good conditioning properties
will be discussed.
Time permitting, the third part of the talk will investigate
p-uniform stability properties of high order Whitney
forms. It will show how tools from differential geometry and
discrete topology can be used to obtain a p-uniform
discrete Poincaré-Friedrichs inequality that plays a
key role in the convergence theory for the p-version
of discrete differential forms.
Anil
N. Hirani
(Control and Dynamical Systems, Caltech) hirani@its.caltech.edu
http://www.cs.caltech.edu/~hirani
Discrete
Exterior Calculus and its Applications in Mechanics and Computer
Science (poster session)
Slides:
pdf
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
Thomas
J.R. Hughes (Institute for Computational Engineering
and Sciences (ICES), The University of Texas at Austin) hughes@ices.utexas.edu
http://www.ticam.utexas.edu/~hughes/
"Consider
a Spherical Cow" - Conservation of Geometry in Analysis: Implications
for Computational Methods in Engineering
"Consider
the spherical cow" is the punch line of a mathematics joke.
The joke itself is not so important to the subject of this work
but the message is, namely, that simplifications of geometry
are often made to facilitate analysis. Let us take the engineering
design process as an example. There are estimated to be of the
order of a million analyses a day performed in engineering design
offices throughout the world. Engineering designs are encapsulated
in Computer Aided Design (CAD) systems. Up to manufacturing
tolerances, these systems exactly represent the geometry of
designs. The analysis process begins with CAD geometry but the
predominate method of analysis, finite elements, requires a
different representation of geometry. This creates two problems:
1) The need to generate the geometric description suitable for
the finite element method; and 2) the geometric errors that
are produced in the process. The first problem, "mesh generation,"
is attributed to taking over 80% of all analysis time in major
engineering industries such as shipbuilding, aerospace and automotive.
It has become the major bottleneck in engineering analysis.
The second problem is very important in certain situations,
such as, for example, the buckling of thin shells, which exhibit
strong geometric imperfection sensitivity. Since approximating
the geometry for analysis purposes is costly, time consuming,
and potentially creates significant errors, it raises the question,
why do we do it? It would seem beneficial to conserve the exact
CAD geometry in analysis, up to, of course, features that we
definitely want to remove. This work takes the point of view
that conserving geometry is an important conservation law that
should be satisfied. We pursue this idea and see where it takes
us. It suggests a very different analytical structure but one
in which mesh generation may be dramatically simplified. Some
simple computations in structural analysis are presented which
indicate the ideas are viable and we argue why we feel that
developing a complete mathematical convergence theory should
be straightforward.
P.
Robert Kotiuga
(Department of Electrical & Computer Engineering, Boston University)
prk@bu.edu
Intuitive
vs. Computable Topological Aspects of Computational Electromagnetics
(poster session)
Slides:
html
pdf
ps
ppt
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.
Melvin
Leok
(Control and Dynamical Systems, California Institute of Technology)
mleok@cds.caltech.edu
Discrete
Connections on Principal Bundles (poster)
Slides:
pdf
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).
Konstantin
Lipnikov (Los
Alamos National Laboratory) lipnikov@t7.lanl.gov
New
Mimetic Discretizations of Diffusion-Type Problems on Polygonal
Meshes (poster session)
Poster
size a0:
pdf
ps
Poster size a4:
pdf
ps
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.
Elizabeth
L. Mansfield (University of Kent, UK) E.L.Mansfield@kent.ac.uk
Towards
a Variational Complex of the Finite Element Method
Slides:
pdf
ps
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).
Ilia
D. Mishev
(Upstream Research Company, Exxon-Mobil) ilya.d.mishev@exxonmobil.com
Why
Mixed Finite Elements are not used in the Petroleum Industry
and what can we do about it? (poster
session)
Slides:
html
pdf
ps
ppt
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.
Jim
E. Morel (Transport
Methods Group, CCS-4, Los Alamos National Laboratory) jim@lanl.gov
Asymptotic-Preserving Discretization Schemes
Slides: pdf
Asymptotic limits associated with partial differential equations
are limits in which certain nondimensional parameters in an
equation are made “small” relative to other nondimensional parameters.
The asymptotic solution is generally found to satisfy an equation
that is much simpler than the original full equation. When the
scale lengths associated with the asymptotic solution are much
larger than the smallest scalelengths associated with the full
equation, it becomes essential from a numerical point of view
to use a discretization scheme for the full equation that “preserves”
the asymptotic limit. An asymptoticpreserving scheme is one
that yields accurate asymptotic solutions whenever the scale
lengths associated with the asymptotic solution are resolved
by the mesh. If a scheme is not asymptotic-preserving, accurate
asymptotic solutions will be obtained only if the smallest scale
lengths associated with the full equation are resolved by the
mesh. Because asymptotic scale lengths can be arbitrarily larger
than the smallest scale lengths of the full equation, this requirement
can make asymptotic calculations prohibitively expensive for
discretization schemes that are not asymptotic-preserving. We
discuss spatial discretization schemes for the radiation transport
equation in the asymptotic diffusion limit. The smallest spatial
scale lengths associated with the transport equation are on
the order of a mean-free-path (the mean-distance between particle
interactions). A truncation error analysis for any consistent
transport spatial discretization scheme will indicate that convergence
to a smooth solution is guaranteed whenever the spatial cell
widths measured in mean-free-paths go to zero. However, the
scale length associated with the diffusion limit can be arbitrarily
large with respect to a mean-free-path. Thus it can be essential
to use asymptotic-preserving discretization schemes in highly
diffusive calculations. We show that the simple upwind scheme
does not preserve the asymptotic diffusion limit, while a discontinuous
Galerkin scheme with a linear trial space does preserve this
limit. Both theory and computational examples are presented.
J.
David Moulton (Los Alamos National Laboratory)
moulton@t7.lanl.gov
Mimetic
Preconditioners for Mixed Discretizations of the Diffusion Equation
(poster session)
Slides:
pdf
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.
Roy
Nicolaides (Department of Mathematical Sciences,
Carnegie Mellon University) nic@cmu.edu
Compatible Discretizations, Covolume Algorithms and
Differential Forms
"Compatible
discretizations" have become prominent during the last few years,
although they have been under development for at least 15 years.
In the finite element setting they are exemplified by "edge"
and related elements. Less well known are "mimetic" and "covolume"
discretizations which may be considered as generalized finite
difference techniques. In most compatible discretizations there
are good analogs of exact sequence diagrams, ensuring the existence
of analogs of vector identities that are valuable for obtaining
good error estimates and reliable numerical results. The covolume
approach uses "complementary volumes" to achieve compatible
discretizations. The complementary volumes are typically tetrahedra
and their corresponding Voronoi polyhedra. Use of these dual
meshes is what distinguishes the covolume methodology from other
compatible discretizations. This talk will begin with a review
of the basic covolume methodology and use it to illustrate the
main ideas of compatible discretization, exact sequences of
spaces and so forth. Following that it will be shown how the
covolume technique can be used to discretize differential forms.
There is a remarkable parallel between certain operations on
differential forms and the basic constructs appearing in covolume
approximations. These will be discussed along with applications
to partial differential equations on manifolds.
Blair Perot
(Department of Engineering & Computer Science, University
of Massachusetts, Amherst) perot@ecs.umass.edu
http://www.ecs.umass.edu/mie/faculty/perot.html
Compatible Reconstructions of Vectors and their Application
to the Navier-Stokes Equations
Slides:
pdf
pps
Compatible spatial discretizations of partial differential equations
invariably deal with vector components as the primary variables.
This use of individual vector components (rather than entire
vectors) is fundamental and closely tied to algebraic topology
and differential forms. However, there are situations where
the entire vector is required. The Navier-Stokes equations for
fluid dynamics exhibit two such situations. First, the advective
term in the equations requires a velocity vector to be defined.
Second, the Navier-Stokes equations have certain vector conservation
statements (like conservation of momentum, kinetic energy, and
circulation) that one would like to construct discrete analogs
for.
Since vectors are not primary quantities, their reconstruction
is a numerical approximation and is not uniquely specified when
compatible spatial discretizations are used. A number of different
vector reconstruction proposals for both structured and unstructured
meshes in 2D and 3D are described. The relationship between
these reconstructions is explored along with the resulting conservation
properties (when they are known to exist). We develop a unified
framework in which vector reconstruction is associated with
discrete averaging operators, and where compatible vector reconstructions
(with conservation statements) have averaging operators which
commute with the discrete differential operators and are closely
related to them.
This work has application even when vectors are not explicitly
required (such as in electromagnetics, elasticity, and Stokes
flow). Discrete Hodge star operators with attractive properties
can be constructed from compatible vector reconstruction operators
(due to the fact that they commute with the differential operators)
even when an explicit vector is never required to solve the
equations.
Ilaria
Perugia (Dipartimento di Matematica, Università
di Pavia) perugia@dimat.unipv.it
http://www-dimat.unipv.it/~perugia
Joint
work with Paul Houston and D.
Schoetzau.
Discontinuous
Galerkin Methods for Maxwell's Equations in Frequency-Domain
(poster session)
Slides:
pdf
We
present recent work on discontinuous Galerkin (DG) methods for
the discretization of the Maxwell equations in frequency-domain,
based on employing the interior penalty discretization of the
curl-curl operator. Direct and mixed methods will be considered
for both low-frequency and high-frequency regimes. The performance
of the proposed DG methods for Maxwell problems will be demonstrated
on a series of model problems with both smooth and singular
analytical solutions. Our theoretical and numerical results
indicate that DG methods provide viable alternatives to classical
conforming edge element discretization schemes.
Robert
N. Rieben
(Institute for Scientific Computing Research, Lawrence Livermore
National Laboratory) rieben1@llnl.gov
http://www.llnl.gov/casc/people/rieben
High
Order Symplectic Integration Methods for Finite Element Solutions
to Time Dependent Maxwell Equations (poster
session)
Slides:
html
pdf
ps
ppt
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.
Beatrice
Riviere (Department of Mathematics, University
of Pittsburgh) riviere@euler.math.pitt.edu
http://www.math.pitt.edu/~riviere
Two-Phase
Flow Modeling (poster session)
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.
Allen
C. Robinson
(Sandia National Laboratories) acrobin@sahp5513.sandia.gov
Compatible
Discretizations in Lagrangian/Eulerian Resistive MHD Modeling
for Z-pinch Applications (poster
session)
Slides:
pdf
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.
Thomas
F. Russell (Division of Mathematical Sciences, National
Science Foundation) thomas.russell@cudenver.edu
http://www-math.cudenver.edu/~trussell/
Relationships Among Some Locally Conservative Discretization
Methods Which Handle Discontinuous Anisotropic Coefficients
on Deformed Grids
Slides:
pdf
This talk presents the relationships between some numerical
methods suitable for a heterogeneous elliptic equation of the
form - div (K(x) grad p) = q, motivated by applications to subsurface
flow (pressure or potential) equations. The methods discussed
are the classical Raviart-Thomas mixed finite element method
(MFEM), the control-volume mixed finite element method (CVMFEM),
the support operators method (SOM), the enhanced cell-centered
finite difference method (ECCFDM), and the multi-point flux-approximation
(MPFA) control volume method. These methods are all locally
mass conservative, and handle general irregular grids with anisotropic
and heterogeneous discontinuous conductivity K(x). In addition
to this, the methods have in common a weak continuity in the
pressure across the edges, which in some cases corresponds to
Lagrange multipliers. This weak continuity appears to be an
essential property for the accuracy of these methods.
While the methods are applicable in two and three dimensions,
the details of the above relationships are presented for logically
rectangular quadrilateral grids in 2D. Issues of deformed grids
are substantially more complex in 3D. An example (Naff, Russell,
and Wilson, Computational Geosciences, 2002) is presented in
which the Piola-transformed lowest-order Raviart-Thomas spaces
fail to contain the constant-velocity vector fields for a hexahedron
that is a trilinear image of a reference cube.
This represents joint work with R.A. Klausen
and R. Winther of the University
of Oslo, Norway.
Rolf
Schuhmann
(Technische Universität Darmstadt, Institut für Theorie Elektromagnetischer
Felder (TEMF)) schuhmann@temf.tu-darmstadt.de
Consistent
Material Operators for Geometrical Discretization Methods on Generalized
Grids (poster session)
Slides:
pdf
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.
Mikhail
Shashkov (Theoretical Division, Los Alamos National
Laboratory) shashkov@lanl.gov
http://cnls.lanl.gov/~shashkov
Mimetic Finite Difference Methods for Partial Differential
Equations and Discrete Vector and Tensor Analysis
Slides:
pdf
In past 10 years we have developed new high-quality, mimetic
finite-difference methods based on discrete analog of vector
and tensor analysis (DVTA). The basis of DVTA is the design
of discrete operators that preserve certain essential properties
of, and relationships between, the corresponding analytic operators.
The DVTA is the basis for new techniques for large-scale numerical
simulations approximating the solution of partial differential
equations (PDEs). The new methods provide a significant extension
of the well known and useful finite volume methods and are designed
to more faithfully represent important properties of physical
processes and the continuum mathematical models of such processes.
Algorithms based on these techniques are used for modeling high-speed
flows, porous media flows, diffusion processes, and electromagnetic
problems. In this presentation we will describe DVTA and demonstrate
how it can be used to construct high-quality finite-difference
methods for PDEs.
Rajen
Kumar Sinha
(Institute for Scientific Computation, Texas A&M University)
rsinha@lechery.isc.tamu.edu
Finite
Volume Element Methods for Parabolic Integro-Differential Equation
with Nonsmooth Initial Data (poster
session)
Paper:
pdf
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
L2 and H1 norms for positive time when
the given initial function is in H01.
Eitan
Tadmor (Center for Scientific Computation And Mathematical
Modeling (CSCAMM), University of Maryland, College Park) tadmor@cscamm.umd.edu
or tadmor@math.umd.edu
http://www.cscamm.umd.edu/~tadmor/
Entropy Stability Theory For Difference Approximations
of Quasilinear Problems
Slides:
pdf
We provide a general overview on the entropy stability of difference
approximations in the context of quasilinear conservation laws,
and related time-dependent problems governed by additional dissipative
and dispersive forcing terms.
As our main tool we use a comparison principle, comparing the
entropy production of a given scheme against properly chosen
entropy-conservative schemes. To this end, we introduce closed-form
expressions for new (families) of new entropy-conservative schemes,
keeping the 'perfect differencing' of the underlying differential
form. In particular, entropy stability is enforced on rarefactions
while keeping sharp resolution of shock discontinuities.
A comparison with the numerical viscosities associated with
these entropy-conservative schemes provides a useful framework
for the construction and analysis of existing and new entropy
stable scheme. We employ this framework for a host of first-
and second-order accurate schemes. The comparison approach yields
precise characterizations of entropy stable semi-discrete schemes
for both scalar problems and multi-dimensional systems of equations.
We extend these results to the fully discrete case, where the
question of stability is settled under optimal CFL conditions
using a complementary approach based on homotopy arguments.
Jean-Marie
Thomas (Applied Mathematics, University of Pau,
France) Jean-Marie.Thomas@univ-pau.fr
Finite
Element Methods for Non-elliptic but Coercive Problems
(poster session)
One
analyze finite element methods for a variational problem : find
u in V such that
a(u,v) - k2 (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) - k2(uh,vh) = l(vh)
for any vh
in Vh ,
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.
Jukka
Tuomela
(Department of Mathematics, University of Joensuu) jukka.tuomela@joensuu.fi
Formal
Theory of PDEs and simulation of Fluid Flows (poster
session)
Poster size a0: imaposteri.pdf
imaposteri.ps
Poster size a4: imaposteria4.pdf imaposteria4.ps
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.
Mary
Fanett Wheeler (Institute for Computational Engineering
and Sciences (ICES), The University of Texas at Austin) mfw@ices.utexas.edu
Coupling of MFE or Mimetic Finite Differences with Discontinuous
Galerkin for Poro-Elasticity
Slides:
html
pdf
ps
ppt
In this presentation we discuss the modeling of Biot's consolidation
model for poro-elasticity in porous media. Here we employ a
mixed finite element or a mimetic finite difference method for
approximating the pore pressure and a discontinuous Galerkin
method for approximating displacements. This approach yields
a locally conservative flow field and continuous numerical fluxes
and forces. Moreover numerical eperiments indicate that the
pore pressure is non-oscillatory even in low permeability regions.
Theoretical error estimates and computational results will be
presented to demonstrate this scheme.
Daniel
A. White (Center for Applied Scientific Computing,
Lawrence Livermore National Laboratory) white37@llnl.gov
http://www.llnl.gov/CASC/people/white/
Development and Application of Compatible Discretizations
of Maxwell's Equations
Slides:
pdf
This talk will be an overview of a variety of electromagnetics
problems that are of interest to LLNL, and of the numerical
methods that have been developed to solve these problems. We
will briefly discuss the history of Finite Difference Time Domain
methods Discrete Surface Integral methods, and Discrete Differential
Form based Finite Element Methods. Our current research efforts
are in the area of higher-order finite elements, higher-order
time integration, and hybrid finite element boundary element
simulations, and we will present recent results in these areas.
Applications that we are applying these algorithms to include:
photonic bandgap structures, linear accelerators, wave propagation
in random environments, and microwave circuits.
This represents joint work with Joe Koning
and Rob Rieben.
Ragnar
Winther (Centre of Mathematic for Applications, University
of Oslo, Norway) rwinther@ifi.uio.no
From the de Rham Sequence to Mixed Elasticity
Slides:
pdf
There have been many attempts during the past four decades to
construct stable mixed finite elements for the Hellinger-Reissner
formulation of linear elasticity, i.e., the stress-displacement
formulation. Unfortunately, these efforts have not been as successful
as expected. There is now renewed interest in this topic due
to applications of mixed models in areas such as viscoelasticity,
where the stress-strain relation may be nonlocal, and, as a
consequence, a pure displacement model is excluded. In this
talk, we first explain why the condition of symmetry of the
stress tensor makes it difficult to construct stable, low order,
mixed finite elements for elasticity. By introducing a proper
commuting diagram, we establish a connection between the standard
de Rham sequence and a corresponding "elasticity sequence."
Utilizing discrete versions of this connection, we are then
able to construct new stable elements in two and three space
dimensions, which satisfy either the usual symmetry condition
or a weak version of this condition.
This represents joint work with Douglas
N. Arnold, Univ. of Minnesota and Richard
S. Falk, Rutgers University.
Ivan
Yotov (Department of Mathematics, University of Pittsburgh)
yotov@math.pitt.edu
http://zeus.math.pitt.edu/~yotov/
Superconvergence in Some Locally Conservative Discretization
Methods
Slides:
pdf
Mixed finite element methods (MFEM), control-volume mixed finite
element methods (CVMFEM), and mimetic finite difference methods
(MFDM) are locally mass conservative discretization methods
that perform well on diffusion-type problems with rough grids
and coefficients. All methods provide accurate approximations
of both the scalar variable (pressure) and its flux (velocity).
Each method has advantages and disadvantages, which will be
discussed. MFEM are naturally formulated as variational methods.
CVMFEM were originally developed as finite volume methods, while
MFDM are based on discrete operators that preserve critical
properties of the differential operators. More recently, CVMFEM
and MFDM have been formulated as variational methods and shown
to be related to MFEM. We discuss how these relationships can
be employed to establish superconvergence for CVMFEM and MFDM
in both pressure and velocity. Extensions of these results to
mortar discretizations on non-matching grids will also be discussed.
This talk reports on joint work with Markus
Berndt, Konstantin Lipnikov,
and Misha Shashkov, LANL, Tom Russell,
University of Colorado Denver, and Mary
Wheeler, University of Texas at Austin.
Photo
Gallery Material
from Talks
Probability
and Statistics in Complex Systems: Genomics, Networks, and Financial
Engineering, September 1, 2003 - June 30, 2004
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