Partial Differential Equations: Modelling and Numerical Simulation (Computational Methods in Applied Sciences)

294 Pages·2008·3.11 MB·English

by  Roland Glowinski

Partial Differential Equations
Computational Methods in Applied Sciences
Volume 16
SeriesEditor
E.Oñate
International CenterforNumericalMethodsinEngineering (CIMNE)
TechnicalUniversityofCatalonia(UPC)
EdificioC-1,CampusNorteUPC
GranCapitán,s/n
08034Barcelona, Spain
[email protected]
www.cimne.com
For other titles published in this series, go to
www.springer.com/series/6899
Partial Differential
Equations
Modeling and Numerical Simulation
Edited by
Roland Glowinski
University ofHouston,TX,USA
and
Pekka Neittaanmäki
UniversityofJyväskylä, Finland
123
Editors
RolandGlowinski PekkaNeittaanmäk i
Department of Mathematics Departmen t of Mathematica lInformation
UniversityofHouston Technology
USA UniversityofJyväs kylä
[email protected] Finland
[email protected]
ISBN978-1-4020-8757-8 e-ISBN978-1-4020-8758-5
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Dedicated to Olivier Pironneau
Preface
For more than 250 years partial differential equations have been clearly the
most important tool available to mankind in order to understand a large
variety of phenomena, natural at first and then those originating from hu-
man activity and technological development. Mechanics, physics and their
engineering applications were the first to benefit from the impact of partial
differential equations on modeling and design, but a little less than a century
agotheSchro¨dingerequationwasthekeyopeningthedoortotheapplication
of partial differential equations to quantum chemistry, for small atomic and
molecular systems at first, but then for systems of fast growing complexity.
The place of partial differential equations in mathematics is a very particular
one: initially, the partial differential equations modeling natural phenomena
were derived by combining calculus with physical reasoning in order to ex-
press conservation laws and principles in partial differential equation form,
leading to the wave equation, the heat equation, the equations of elasticity,
the Euler and Navier–Stokes equations for fluids, the Maxwell equations of
electro-magnetics,etc.Itisinordertosolve‘constructively’theheatequation
that Fourier developed the series bearing his name in the early 19th century;
Fourier series (and later integrals) have played (and still play) a fundamental
roleinbothpureandappliedmathematics,includingmanyareasquiteremote
from partial differential equations.
Ontheotherhand,severalareasofmathematicssuchasdifferentialgeom-
etry have benefited from their interactions with partial differential equations.
The need for a better understanding of the properties of the solution of these
equations has been a driver for both the mathematical investigation of their
existence, uniqueness, regularity, and other properties, and the development
of constructive methods to approximate these solutions. Numerical methods
for the approximate solution of partial differential equations were invented,
developed and applied to real life situations long before the advance (in the
mid-forties) of digital computers; let us mention among these early methods:
finite differences, Galerkin, Courant finite element, and a variety of iterative
methods. However, the exponential growth in speed and memory of digital
VIII Preface
computers has been at the origin of an explosive development of numerical
mathematics,leadingitselftoapplicationsofsizeandcomplexityunthinkable
a not so long time ago.
There has been simultaneity in the progress achieved on both the theory
and the numerics of partial differential equations, each feeding the other one:
indeed, methods for proving the existence of solutions have lead to numerical
methods for the actual computation of these solutions; on the other hand,
conjectures on mathematical properties of solutions have been verified first
computationallyprovidingthusajustificationforfurtheranalyticalinvestiga-
tions. Applications of partial differential equations are essentially everywhere
since to the areas mentioned above we have to add bio and health sciences,
finance,imageprocessing.(Itisworthmentioningthattodaythetermpartial
differentialequationshastobetakeninabroadersensethanletsayfiftyyears
agoinordertoincludepartialdifferentialinequalities,whichareoffundamen-
tal importance in, for example, the modeling of non-smooth phenomena.)
From the above comments, it is quite obvious that the “world of partial
differential equations” is a very large and complex one, and, therefore, quite
difficult to explore. Not surprisingly, the many aspects of partial differential
equations(theory,modelingandcomputation)havemotivatedahugenumber
ofpublications(books,articles,conferenceproceedings,websites).Concerning
books, most of them are necessarily specialized (unless elementary) with top-
ics such as elliptic equations, parabolic equations, Navier–Stokes equations,
Maxwell equations, to name some of the most popular ones. We think thus
that thereis aneed for books on partial differential equations addressingata
reasonably advanced level a variety of topics. From a practical point of view,
the diversity we mentioned above implies that such books have to be neces-
sarilymulti-authors.Wethinkthatthepresentvolumeisananswertosucha
need since it contains the contributions of experts of international reputation
on a quite diverse selection of topics all partial differential equation related,
ranging from well-established ones in mechanics and physics to very recent
ones in micro-electronics and finance. In all these contributions the emphasis
has been on the modeling and computational aspects.
Thisvolumeisstructuredasfollows:InPartI,discontinuousGalerkinand
mixed finite element methods are applied to a variety of linear and nonlinear
problems, including the Stokes problem from fluid mechanics and fully non-
linearellipticequationsoftheMonge-Amp`eretype.PartIIisdedicatedtothe
numericalsolutionoflinearandnonlinearhyperbolicproblems.InPartIIIone
discusses the solution by domain decomposition methods of scattering prob-
lems for wave models and of electronic structure related nonlinear variational
problems. Part IV is devoted to various issues concerning the modeling and
simulation of fluid mechanics phenomena involving free surfaces and moving
boundaries.Thefinitedifferencesolutionofaproblemfromspectralgeometry
has also been included in this part. Part V is dedicated to inverse problems.
Finally, in Part VI one addresses the parabolic variational inequalities based
modeling and simulation of finance related processes.
Preface IX
Some of the issues discussed in this volume have been addressed at the
international conference taking place in Helsinki during fall 2005 to honor
OlivierPironneauontheoccasionofhis60thanniversary.Additionalmaterial
has been included in order to broaden the scope of the volume.
Special acknowledgements are due to Marja-Leena Rantalainen from Uni-
versityofJyva¨skyla¨forhermostconstructiveroleinthevariousstagesofthis
project.
Houston and Jyva¨skyla¨ Roland Glowinski
Pekka Neittaanm¨aki
Contents
List of Contributors .......................................... XIII
Part I Discontinuous Galerkin and Mixed Finite Element Methods
Discontinuous Galerkin Methods
Vivette Girault and Mary F. Wheeler.............................. 3
Mixed Finite Element Methods on Polyhedral Meshes
for Diffusion Equations
Yuri A. Kuznetsov .............................................. 27
On the Numerical Solution of the Elliptic Monge–Amp`ere
Equation in Dimension Two: A Least-Squares Approach
Edward J. Dean and Roland Glowinski............................. 43
Part II Linear and Nonlinear Hyperbolic Problems
Higher Order Time Stepping for Second Order Hyperbolic
Problems and Optimal CFL Conditions
J. Charles Gilbert and Patrick Joly................................ 67
Comparison of Two Explicit Time Domain Unstructured
Mesh Algorithms for Computational Electromagnetics
Igor Sazonov, Oubay Hassan, Ken Morgan, and Nigel P. Weatherill ... 95
The von Neumann Triple Point Paradox
Richard Sanders and Allen M. Tesdall ............................. 113
Part III Domain Decomposition Methods
ALagrangeMultiplierBasedDomainDecompositionMethodfor
theSolutionofaWaveProblemwithDiscontinuousCoefficients
Serguei Lapin, Alexander Lapin, Jacques P´eriaux,
and Pierre-Marie Jacquart ....................................... 131