Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems

328 Pages·2013·3.391 MB·English

by  Clemens Pechstein (auth.)

Lecture Notes
in Computational Science 90
and Engineering
Editors:
TimothyJ. Barth
MichaelGriebel
DavidE. Keyes
RistoM. Nieminen
DirkRoose
TamarSchlick
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Clemens Pechstein
Finite and Boundary
Element Tearing
and Interconnecting Solvers
for Multiscale Problems
123
ClemensPechstein
InstituteofComputationalMathematics
JohannesKeplerUniversity
Linz
Austria
ISSN1439-7358
ISBN978-3-642-23587-0 ISBN978-3-642-23588-7(eBook)
DOI10.1007/978-3-642-23588-7
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012953096
Math.Subj.Class.(2010):65F08,65N22,65N30,65N38,65N55,65Y05
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Preface
The purpose of this book is to give a detailed and self-contained presentation of
tearingandinterconnectingmethodsforfiniteandboundaryelementdiscretizations
of elliptic partial differential equations. This includes a description of the corre-
sponding algorithms as well as a rigorous convergence theory. In particular, two
issuesareaddressedwhichhavenotbeentreatedinanypreviousmonograph.Firstly,
weconsideralsothecaseofboundaryelementtearingandinterconnecting(BETI),
thecouplingoffiniteandboundaryelementswithinthetearingandinterconnecting
framework, and the application to exterior problems. Secondly, we consider the
case of highly varying (multiscale) coefficients not resolved by the subdomain
partitioning. In this sense, this book serves as a good complementto the existing
monographsandsurveysbyFarhatandRoux[FR94],ToselliandWidlund[TW05],
Korneev and Langer [KL04], and Mathew [Mat08], which already treat the finite
elementtearingandinterconnecting(FETI)method,dual-primalFETI(FETI-DP),
andbalancingdomaindecompositionbyconstraints(BDDC).
As model problem throughout, we use the well-known scalar elliptic problem
(cid:2)div.˛ru/Df, discretized by continuous piecewise linear finite elements or
boundary elements. Extensions to other PDEs and other discretizations will be
discussedbriefly.
Tearing and interconnecting methods are special non-overlapping domain
decompositionmethods.Theycanbeperceivedas
(i) Couplingmethodsbetweenpossiblydifferentdiscretizations,and/or
(ii) Iterativesolversforlarge-scaleequationswell-suitedforparallelization.
The main idea of tearing and interconnectingmethodsis to subdividethe compu-
tational domain into non-overlapping subdomains. The equation is posed locally
on each subdomains, and the coupling of these local problems is performed by
Lagrangian multipliers. This procedure corresponds to aspect (i). Using different
strategies, the original variables can be eliminated, and the remaining equation
involvesonlytheLagrangianmultipliers.Tosolvethis“dual”problem,oneusually
employs an iterative method with a certain preconditioner. In many cases, the
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viii Preface
resultingparallelmethodisquasi-optimal,i.e.,thetotalcomputationaleffortgrows
onlypolylogarithmicallyinthenumberofunknownspersubdomain.
We will not elaborate the general subject of domain decomposition methods
(butreferto[Mat08,SBG96,QV99,TW05]).However,toensureaself-contained
presentation,wehavecollectedthebasicsofSobolevspaces,projections,finiteand
boundary elements, Schur complements, etc. into Chap.1. The remainder of the
bookisorganizedasfollows.
Chapter2discussesindetailtheformulationof(one-level)FETI/BETImethods.
Besides the classical formulation,we also consider the total FETI method and its
boundary element counterpart, all-floating BETI, as well as the closely related
balancing Neumann-Neumannmethods. Furthermore, we provide a rigorouscon-
vergenceanalysisforthemodelproblemundercertaingeometricassumptionsand
ontheassumptionthatthecoefficient˛isconstantineachsubdomain.
ThescopeofChap.3istorelaxtheassumptionson˛ andallowformorerapid
or noisy variation, not necessarily resolved by the subdomains. Note that precon-
ditioning such multiscale problems is in itself quite an old discipline. However,
manyearlierworksassumethatthecoefficientisresolvedbythesubdomainsorthe
coarsemesh.Thebranchofresearchthatconsiderscoefficientsbeingnotresolved
bythesubdomains/coarsemeshisratheryoungandgrowing(seethereferencesin
Sect.3.1.1).Inthiscontext,thebookoffersadetailedviewtoanactiveandup-to-
dateareaofresearch.Themaintoolinthischapter,aweightedPoincare´ inequality
(WPI),willbediscussedindetail.
Chapter 4 is devoted to exterior problems. Here, one of the subdomains is
unboundedandtreatedbyboundaryelements.Wediscusshowandtowhichextent
the theory of Chap.2 can be generalized. Due to the fact that the unbounded
subdomain can have many neighboring subdomains, there are new difficulties
comparedtotheboundedcase,whichhavetobetreatedsuitably.
In Chap.5, we formulate the dual-primal FETI/BETI methods and the closely
related BDDC methods. As in Chap. 2, we treat both theoretical and algorithmic
aspects. We also discuss extensions to the unbounded case from Chap.4, and to
someextenttothemultiscaleproblemsfromChap.3.
Somecommentsonthebibliography.Ihavetriedtoprovidethebasicandoriginal
referencesinthefield,asup-to-dateaspossible.However,thereareprobablysome
more(beyondthe300)thatIhavemissed,andIapologizeforthis.
Thanksareduetoseveralpeople.MydearparentsBarbaraandGerdgavemethe
opportunityof high-leveleducationatschoolanduniversity.UlrichLangertaught
me numerical analysis, supervised my doctoral thesis, and he encouraged me to
write this monograph. My collaborator and friend Robert Scheichl invested his
time and energy for our joint research, motivated and pushed me. I have profited
tremendously from numerous scientific discussions with Clark Dohrmann, Axel
Klawonn,Gu¨ntherOf,OliverRheinbach,MarcusSarkis,OlafSteinbach,andOlof
Widlund. Helpful remarks on the manuscript were provided by Rob Scheichl,
Clark Dohrmann,Clemens Hofreither,and Bedˇrich Soused´ık.At Springer-Verlag,
Martin Peters and Thanh-Ha Le Thi took care about the organization around
this monograph; thanks also for their patience. My colleagues at the Institute of
Preface ix
Computational Mathematics and the associated group of the Radon Institute for
Computational and Applied Mathematics (RICAM) at Linz spent time to discuss
now and then or to cheer me up, especially Walter Zulehner. It should also be
mentionedthattheAustrianScienceFund(FWF)supportedalotofmyresearch.
Finally, there is my wonderful wife Astrid, without whom this monograph
wouldhaveneverbeenfinished.Notonlydidshesupportmewithlove,faith,and
understanding,but she providedenoughpressure (to completethe book)and read
severaltimes throughthe manuscript!Moreover,she gavebirthto the sunshineof
ourlife,ourlovelydaughterJohanna.
Linz,Austria ClemensPechstein
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