Table Of ContentTo Krystyna and Katja
Love’s not Time’s fool
— William Shakespeare, Sonnett 116
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Preface
Als u¨berragende Gestalt ...tritt uns Helmholtz entgegen ...Seine
außerordentliche Stellung in der Geschichte der Naturwissenschaf-
tenberuhtaufeinerungew¨ohnlichvielseitigen,eindringendenBega-
bung,innerhalbderendiemathematischeSeiteeinewichtige,fu¨runs
natu¨rlich in erster Linie in Betracht kommende Rolle spielt. (Felix
Klein, [84, p. 223])1
Waves are interesting physical phenomena with important practical appli-
cations. Physicists and engineers are interested in the reliable simulation
of processes in which waves are scattered from obstacles (scattering prob-
lems). This book deals with some of the mathematical issues arising in the
computational simulation of wave propagation and fluid–structure interac-
tion.
Thelinearmathematicalmodelsforwavepropagationandscatteringare
well-known. Assuming time-harmonic behavior, one deals with the Helm-
holtz equation ∆u + k2u = 0, where the wave number k is a physical
parameter. Our interest will be mainly in the numerical solution of exte-
rior boundary value problems for the Helmholtz equation which we call
Helmholtz problems for short.
The Helmholtz equation belongs to the classical equations of mathema-
tical physics. The fundamental questions about existence and uniqueness
1InHelmholtzwemeetanoverwhelmingpersonality.Hisextraordinarypositionin
thehistoryofscienceisbasedonhisunusuallydiverseandpenetratingtalents,among
which the mathematical side, which for our present purpose is of primary importance,
playsanimportantrole.
viii Preface
ofsolutionstoHelmholtzproblemsweresolvedbytheendofthe1950s;cf.,
e.g., the monographs of Leis [87], Colton–Kress [39], and Sanchez Hubert–
Sanchez Palencia [107]. Those results of mathematical analysis form the
fundamental layer on which the numerical analysis in this book is built.
The two main topics that are discussed here arise from the practical app-
lication of finite element methods (FEM) to Helmholtz problems.
First,FEMhavebeenconceptuallydevelopedforthenumericaldiscreti-
zation of problems on bounded domains. Their application to unbounded
domainsinvolvesadomaindecompositionbyintroducinganartificialboun-
daryaroundtheobstacle.Attheartificialboundary,thefiniteelementdis-
cretization can be coupled in various ways to some discrete representation
of the analytical solution. We review some of the coupling approaches in
Chapter3,focusingonthosemethodsthatarebasedontheseriesrepresen-
tation of the exterior solution. In particular, we review localized Dirichlet-
to-Neumannandotherabsorbingboundaryconditions,aswellastherecent
perfectly matched layer method and infinite elements.
Second,whenusingdiscretemethodsforthesolutionofHelmholtzprob-
lems, one soon is confronted with the significance of the parameter k. The
wave number characterizes the oscillatory behavior of the exact solution.
The larger the value of k, the stronger the oscillations. This feature has to
be resolved by the numerical model. The “rule of thumb” is to resolve a
wavelength by a certain fixed number of mesh points. It has been known
from computational experience that this rule is not sufficient to obtain re-
liable results for large k. Looking at this problem from the viewpoint of
numerical analysis, the reason for the defect can be found in the loss of
operator stability at large wave numbers. We address this topic in Chap-
ter 4, where we present new estimates that precisely characterize the error
behaviorintherangeofengineeringcomputations.Wecalltheseestimates
preasymptotic in order to distinguish them from the well-known asympto-
tic error estimates for indefinite problems satisfying a G˚arding inequality.
In particular, we accentuate the advantages of the hp-version of the FEM,
as opposed to piecewise linear approximation. We also touch upon gene-
ralized (stabilized) FEM and investigate a posteriori error estimation for
Helmholtz problems. Our theoretical results are obtained mainly for one-
dimensionalmodelproblemsthatdisplaymostoftheessentialfeaturesthat
matter in the true simulations.
In the introductory Chapters 1 and 2, we set the stage for the finite ele-
ment analysis. We start with an outline of the governing equations. While
our physical application is acoustic fluid–structure interaction, much of
the mathematics in this book may be relevant also for numerical electro-
dynamics. We therefore include a short section on Maxwell’s equations.
In Chapter 2, we first (Section 2.1) review mathematical techniques for
the analytical solution of exterior Helmholtz problems. Our focus is on the
separation of variables and series representations of the solution (comple-
mentary to the integral methods and representations), as needed for the
Preface ix
outline of the coupling methods in Chapter 3. The second part (Sections
2.2–2.5) of Chapter 2 is a preparation of the finite element analysis in
Chapter 4. We first briefly review some necessary definitions and theorems
from functional analysis inasmuch as they are needed for the subsequent
investigation. Then we consider the variational formulation of Helmholtz
problems and discuss variational methods.
Computational results for three-dimensional scattering problems are re-
ported in Chapter 5.
This text is addressed to mathematicians as well as to physicists and
computational engineers working on scattering problems. Having a mixed
audience in mind, we attempted to make the text self-contained and easily
readable.ThisespeciallyconcernsChapters3and4.Wehopethattheillus-
trationwithmanynumericalexamplesmakesforabetterunderstandingof
thetheory.Thematerialoftheintroductorychaptersispresentedinamore
compact manner for the convenience of later reference. It is assumed that
the reader is familiar with the basic physical and mathematical concepts
of fluid–structure interaction and/or finite element analysis. References to
various expositions of these topics are supplied.
Acknowledgments
Es ist eben viel wichtiger, in welche geistige Umgebung ein Mensch
hineinkommt, die ihn viel st¨arker beeinflußt als Tatsachen und kon-
kretes Wissen, das ihm geboten wird. (Felix Klein, [84, p. 249])2
Much of this book is a report of my own cognitive journey towards the
reliablesimulationofscatteringproblemswithfiniteelementmethods.My
interestinnumericalacousticsbeganwhileIwasworkingasanassociateof
IvoBabuˇskaattheUniversityofMarylandatCollegePark.Manyresultsin
thisbookstemfromourjointwork,andIhavetriedmybesttoputthespir-
it of our discussions down on paper. My gratitude goes to J. Tinsley Oden
and Leszek Demkowicz, of the Texas Institute for Computational and Ap-
plied Mathematics (TICAM). Most of this monograph was written during
my appointment as a TICAM Research Fellow, and TICAM’s extraordi-
nary working conditions and stimulating intellectual atmosphere were an
essential ingredient for its shape and content. I gratefully acknowledge the
financial support from the Deutscher Akademischer Austauschdienst and
theDeutscheForschungsgemeinschaft.ThankstoProfessorsReißmannand
R¨ohr from my home University of Rostock, Germany, for their support of
my grant applications. With deep gratitude I acknowledge the close co-
operationwithJosephShirronoftheNavalResearchLaboratory(NRL)in
Washington,D.C.,whomadehisprogramSONAXavailabletomeandwas
alwaysreadytosharehisbroadandsolidexperienceinnumericalacoustics.
2Theintellectualenvironmentapersonentersismoresignificantandwillbeofmuch
greaterinfluencethanthefactsandconcreteknowledgethatareofferedhim.
