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二维外部问题的混合谱方法(英文).docx

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二维外部问题的混合谱方法(英文) MixedSpectralMethodsforTwo-DimensionalExteriorProblems Introduction: Thestudyofspectralmethodshasbecomeincreasinglypopularoverthepastfewdecades.Spectralmethodsarenumericaltechniquesforsolvingpartialdifferentialequations(PDEs)andotherproblems.Thesemethodsareparticularlyeffectiveforproblemsinwhichthesolutionshavewell-definedspectraorfrequencypatterns.Spectralmethodsareoftenusedintwo-dimensionalproblemsthatinvolveexteriordomains,whichareregionsthatareboundedononesideandunboundedontheother. TheoreticalBackground: Themixedspectralmethodisacombinationoffiniteelementandspectralmethods.ThismethodinvolvesexpandingthesolutionofaPDEintermsofafinitenumberofbasisfunctions,whicharetypicallychosentobeorthogonalandwell-behaved.Thesebasisfunctionscanbechoseninavarietyofways,includingB-splinebasisfunctions,Chebyshevpolynomials,andLegendrepolynomials. ThespectralmethodinvolvesapproximatingthesolutionofaPDEbyatruncatedseriesexpansionintermsofasetofwell-chosenbasisfunctions.Thesebasisfunctionsareoftenchosentobeorthogonal,andwell-behaved.CommonexamplesofbasisfunctionsusedinspectralmethodsincludeChebyshevpolynomials,Legendrepolynomials,andFourierseries. Thecombinedfiniteelementandspectralmethodhelpstoovercomethelimitationsofeachindividualmethodindealingwithcomplexgeometries.Finiteelementmethodsarewell-suitedforsolvingproblemsincomplexgeometriesbecausetheyuseameshtodiscretizetheregionofinterest.However,thesemethodscansufferfromnumericalinstabilitiesandconvergenceissues.Spectralmethods,ontheotherhand,donotrequireameshandcanbeusedtosolveproblemsinmoregeneralgeometries.However,spectralmethodscanbemorecomputationallyexpensiveandrequiremoreadvancedmathematicalknowledgetoimplement. MixedSpectralMethodsforExteriorProblems: Mixedspectralmethodshavebeensuccessfullyappliedtoavarietyoftwo-dimensionalexteriorproblems.Someexamplesoftheseproblemsincludewavescattering,fluiddynamics,andelectromagnetictheory. Inwavescatteringproblems,mixedspectralmethodshavebeenusedtocomputethescatteringof