A multigrid method for nonlinear unstructured finite element

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UCRL-JC- 150513

A Multigrid Method for

Nonlinear Unstructured

Finite Element Elliptic

Equations

Miguel A. Dumett, Panayot Vassilevski, and Carol S.

Woodward.

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SIAM Journal on Scientific Computing

October 2002

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AMULTIGRIDMETHODFORNONLINEARUNSTRUCTURED

FINITEELEMENTELLIPTICEQUATIONS

MIGUELA.DUMETT,PANAYOTS.VASSILEVSKI,ANDCAROLS.WOODWARD Abstract.Thispaperpresentsanapplicationoftheelementagglomeration-basedcoarsen-ingprocedure(agglomerationAMGe)proposedin[10],tobuildthecomponentsofamultigridmethodforsolvingnonlinear niteelementellipticequationsongeneralunstructuredmeshes.Theagglomeration-basedAMGeo erstheabilitytode necoarseelementsandelementmatrices,pro-videdaccesstoelementsandelementmatricesonthe negridisavailable.Wefocusontheper-formanceoftheclassicalfullapproximationscheme(FAS).Inthepresentcontextthecoarsenodesareconstructedalgebraicallybasedontheelementagglomeration,andtheinterpolationrulesarebasedonthe(linear)AMGeexploitingelementmatricesofLaplaceoperatorandL2-masselementmatrices.TheAMGeprovidesthecoarsecounterpartsonalllevels.Thenonlinearcoe cientsareaveragedoverthecoarseelements,whichleadstonon-inheritedformsandhencetonon-inheritedmultigridmethods.Numericalresultsshowthattheresultingnonlinearmultigridgivesmeshinde-pendentconvergenceonmodelproblems.Inaddition,thenonlinearmultigridschemeappearstobemoree cientandrobustforpoorinitialguessesthanrepeatedapplicationsofthenonlinearsystemsmoother(i.e.,singlelevelmethod).Finally,ournumericalresultsindicatethathandlingnonlinear-itiesoncoarsegridscanprovideanadvantageovernonlinearsolversthathandlenonlinearitiesonlyontheoriginalproblemgrid.

Keywords.:nonlinearellipticequations,unstructuredmeshes, niteelements,FAS,algebraicmultigrid,inexactNewton,Picard,AMGe.

AMSsubjectclassi cations.:65N30,65N55.

1.Introduction.Thispaperaddressesnonlinearellipticequationsdiscretizedongenerallyunstructuredmeshesusing niteelements.Theunstructured niteele-mentsarewidelyusedinpracticebecauseoftheirbetteradaptationtogeometricalorcoe cientirregularitiesoftheellipticoperator.Therefore,unstructuredmeshesarenotgenerallyobtainedbysuccessivestepsofre nement;coarsemeshes,ifneeded,mustbegeneratedalgebraically.Weintendtoinvestigatetheperformanceoftheclas-sicalfullapproximationscheme(FAS)[1]appliedtothespeci edclassofnonlinearellipticequations.WewillcompareFASwithsomestandardnonlinearschemes,likeinexactNewtonandPicard.Thispapershouldbeviewedasapreliminary rststudyonthissubjecttoassessthepotentialofthedevelopednonlinearmultigridmethod.Moredetailedandsophisticatedcomparisonisyettobeperformed.Theapproachofgeneratingcoarsenonlinearproblemstakeninthispapercanbeviewedasanex-tensionoftheelementagglomerationAMGeproposedin[10].Thisextensionofthe(linear)agglomeration-basedAMGeprovidesallcomponentsforanonlinearmulti-grid(coarsegrids,interpolationrules,andcoarsenonlinearoperators).TheresultingFASalgorithmwillbereferredtohereinastheFAS-AMGemethod.Apreliminaryversionofthemethodwasreportedin[11].NotethatourmethodsofextendingtheAMGeframeworkarealsorelevantforthenonlinearmultigridmethod(NMGM)ofHackbusch[8].Extendingourworktothismethodisapossibletopicoffuturework.

Onstructuredgrids,thecoarseningstrategy,thegridtransferoperatorsandthecoarsegridoperator(neededbyanymultigridmethod)canbede nedinastraightfor-wardgeometricwayforbothlinearandnonlinearmultigridalgorithms.Thehierarchy

workwasperformedundertheauspicesoftheU.S.DepartmentofEnergybyUniversityofCaliforniaLawrenceLivermoreNationalLaboratoryundercontractnumberW-7405-Eng-48. CenterforAppliedScienti cComputing,LawrenceLivermoreNationalLaboratory,Livermore,CA94551;(dumett1@llnl.gov,panayot@llnl.gov,cswoodward@llnl.gov).

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