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1、FiniteElementAnalysisofAcousticScatteringFrankIhlenburgSpringerToKrystynaandKatjaLove’snotTime’sfool—WilliamShakespeare,Sonnett116ThispageintentionallyleftblankPrefaceAls¨uberragendeGestalt...trittunsHelmholtzentgegen...SeineaußerordentlicheStellunginderGeschichtederNaturwi
2、ssenschaf-tenberuhtaufeinerungew¨ohnlichvielseitigen,eindringendenBega-bung,innerhalbderendiemathematischeSeiteeinewichtige,f¨urunsnat¨urlichinersterLinieinBetrachtkommendeRollespielt.(Felix1Klein,[84,p.223])Wavesareinterestingphysicalphenomenawithimportantpracticalappli-ca
3、tions.Physicistsandengineersareinterestedinthereliablesimulationofprocessesinwhichwavesarescatteredfromobstacles(scatteringprob-lems).Thisbookdealswithsomeofthemathematicalissuesarisinginthecomputationalsimulationofwavepropagationandfluid–structureinterac-tion.Thelinearmathe
4、maticalmodelsforwavepropagationandscatteringarewell-known.Assumingtime-harmonicbehavior,onedealswiththeHelm-holtzequation∆u+k2u=0,wherethewavenumberkisaphysicalparameter.Ourinterestwillbemainlyinthenumericalsolutionofexte-riorboundaryvalueproblemsfortheHelmholtzequationwhic
5、hwecallHelmholtzproblemsforshort.TheHelmholtzequationbelongstotheclassicalequationsofmathema-ticalphysics.Thefundamentalquestionsaboutexistenceanduniqueness1InHelmholtzwemeetanoverwhelmingpersonality.Hisextraordinarypositioninthehistoryofscienceisbasedonhisunusuallydiversea
6、ndpenetratingtalents,amongwhichthemathematicalside,whichforourpresentpurposeisofprimaryimportance,playsanimportantrole.viiiPrefaceofsolutionstoHelmholtzproblemsweresolvedbytheendofthe1950s;cf.,e.g.,themonographsofLeis[87],Colton–Kress[39],andSanchezHubert–SanchezPalencia[10
7、7].Thoseresultsofmathematicalanalysisformthefundamentallayeronwhichthenumericalanalysisinthisbookisbuilt.Thetwomaintopicsthatarediscussedherearisefromthepracticalapp-licationoffiniteelementmethods(FEM)toHelmholtzproblems.First,FEMhavebeenconceptuallydevelopedforthenumericald
8、iscreti-zationofproblemsonboundeddomains.Theirapplicationtounboundeddomainsinvolve
