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1、ASimpleMultigridSchemeforSolvingthePoissonEquationwithArbitraryDomainBoundaries.ThomasGuilleta,RomainTeyssiera,baIRFU/SAp,CEASaclay,Bˆatiment709,91191Gif-sur-YvetteCedex,FrancebInstituteofTheoreticalPhysics,UniversityofZ¨urich,Winterthurerstrasse190,8057Z¨urich,SwitzerlandAbstractWep
2、resentanewmultigridschemeforsolvingthePoissonequationwithDirichletboundaryconditionsonaCartesiangridwithirregulardomainboundaries.ThisschemewasdevelopedinthecontextoftheAdaptiveMeshRefinement(AMR)schemesbasedonagraded-octreedatastructure.ThePoissonequationissolvedonalevel-by-levelbasi
3、s,usinga“one-wayinter-face”schemeinwhichboundaryconditionsareinterpolatedfromtheprevi-ouscoarserlevelsolution.Suchaschemeisparticularlywellsuitedforself-gravitatingastrophysicalflowsrequiringanadaptivetimesteppingstrategy.ByconstructingamultigridhierarchycoveringtheactivecellsofeachAM
4、Rlevel,wehavedesignedamemory-efficientalgorithmthatcanbenefitfullyfromthemultigridacceleration.Wepresentasimplemethodforcapturingtheboundaryconditionsacrossthemultigridhierarchy,basedonasecond-orderaccuratereconstructionoftheboundariesofthemultigridlevels.Incaseofverycomplexboundaries,s
5、mallscalefeaturesbecomesmallerthanthediscretizationcellsizeofcoarsemultigridlevelsandconvergenceprob-lemsarise.Weproposeasimplesolutiontoaddresstheseissues.Usingourscheme,theconvergencerateusuallydependsonthegridsizeforcom-plexgrids,butgoodlinearconvergenceismaintained.Theproposedmet
6、hodwassuccessfullyimplementedondistributedmemoryarchitecturesintheRAMSEScode,forwhichwepresentanddiscussconvergenceandaccuracyarXiv:1104.1703v1[physics.comp-ph]9Apr2011propertiesaswellastimingperformances.Keywords:Poissonequation;Multigridmethods;Adaptivemeshrefinement;Ellipticmethods
7、PreprintsubmittedtoComputationalPhysicsApril12,20111.IntroductionEllipticalpartialdifferentialequationsplayacentralroleinmanymath-ematicalandphysicalproblems.ThePoissonequation,inparticular,arisesnaturallyinthecontextofelectromagnetism,fluiddynamicsandgravity.Itisthereforeofgreatsignifi
8、canceinastro
