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Mathematical Models and Methods in Applied Sciences fc World Scientific Publishing Company

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Niveau: Supérieur, Doctorat, Bac+8
Mathematical Models and Methods in Applied Sciences fc World Scientific Publishing Company Mortaring the two-dimensional edge finite elements for the discretization of some electromagnetic models? Adnene BEN ABDALLAH Applications Scientifiques pour le Calcul Intensif, UPR 9069 CNRS, Bat. 506, Universite Paris Sud, 91405 Orsay, France. Faker BEN BELGACEM Mathematiques pour l'Industrie et la Physique, UMR 5640 CNRS, Univ. Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France. Yvon MADAY Laboratoire J.-L. Lions, UMR 7583 CNRS, Univ. Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France. Francesca RAPETTI Laboratoire J.-A. Dieudonne, UMR 6621 CNRS, Univ. Nice et Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 02, France. We describe the mortar element method for the two-dimensional edge finite elements of class H(curl). These finite elements are currently used for the discretization of various models coming from the Maxwell equations and using them in a mortar framework has several interesting applications in the electromagnetic and electrotechnical domains. We develop some technical tools necessary to perform the numerical analysis of this approach. Then, we prove some optimal approximation results and we illustrate the theory by some numerical experiences.

  • netic wave

  • through translations

  • equation similar

  • ?n ?

  • time discretization

  • tangential component operator

  • dirichlet boundary

  • propagation domain

  • treated through numerical


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1Wedescribethemortarelementmethodforthetwo-dimensionaledgefiniteelementsofclassH(curl).ThesefiniteelementsarecurrentlyusedforthediscretizationofvariousmodelscomingfromtheMaxwellequationsandusingtheminamortarframeworkhasseveralinterestingapplicationsintheelectromagneticandelectrotechnicaldomains.Wedevelopsometechnicaltoolsnecessarytoperformthenumericalanalysisofthisapproach.Then,weprovesomeoptimalapproximationresultsandweillustratethetheorybysomenumericalexperiences.Keywords:edgefiniteelements,non-matchinggrids,mortarelementmethod1.IntroductionDomaindecompositiontechniqueshavebecomeastandardwaytoincreasethesizeoftheproblemsthatcanbetreatedthroughnumericalsimulation.ThesetechniquesTheauthorsrecallthat,despiteitslatepublication,thispaperisthestartingsteptowardtheapplicationofthemortarelementmethodtoelectromagnetism.Mortaringthetwo-dimensionaledgefiniteelementsforthediscretizationofsomeelectromagneticmodelsFrancescaRAPETTILaboratoireJ.-A.Dieudonne´,UMR6621CNRS,Univ.NiceetSophia-Antipolis,ParcValrose,06108Nicecedex02,France.frapetti@math.unice.frAdne`neBENABDALLAHApplicationsScientifiquespourleCalculIntensif,UPR9069CNRS,Baˆt.506,Universite´ParisSud,91405Orsay,France.adnene@asci.frYvonMADAYLaboratoireJ.-L.Lions,UMR7583CNRS,Univ.PierreetMarieCurie,4placeJussieu,75252Pariscedex05,France.maday@ann.jussieu.frFakerBENBELGACEMMathe´matiquespourl’IndustrieetlaPhysique,UMR5640CNRS,Univ.PaulSabatier,118routedeNarbonne,31062ToulouseCedex04,France.belgacem@mip-tlse.frynapmoCgnihsilbuPcitneicSdlroWcsecneicSdeilppAnisdohteMdnasledoMlacitamehtaM
2A.BenAbdallah,F.BenBelgacem,Y.Maday,F.Rapettiallowtouseefficientlytheparallelpotentialityofcomputersandtodesignrobustiterativesolverstoacceleratetheaccesstothenumerical(approximated)solution.Whendealingwithellipticorparabolicproblems,inordertoincreaseevenmoretheflexibilityofthedomaindecompositionapproach,themortarelementmethod19,10allowstouse,ineachsub-domainofthedecomposition,theproperdiscretiza-tion(offiniteelement,spectral,finitevolumeorofwavelettype)wellsuitedtothelocal(relatedtothesub-domain)solutionbehaviorandgeometricalfeatures.Theresultingapproximationisoptimalintermsoferrorestimatesandconvergenceratessincethelocaldiscretizationparameterscanbechosenindependentlyovereachsub-domaininsuchawaythateachlocalcontributiontotheerrorisbalanced.Thisdiscretizationparametercaneitherbedesignedapriorioraposteriori42,18inasuitableway.Themortarelementmethodhasreceivedinthelastdecadealotofattentionanditiscurrentlyusedinmanyareaofthenumericalsimulationsuchas,forex-ample,forthespectralapproximationofNavierStokesequations(intheprimalvariablesformulation6,12,andintheψor(ψ,ω)formulation8),forthefiniteele-mentapproximationofNavierStokesequations(intheprimalvariablesformulation3,11),forthefiniteelementapproachofelasticity37,36.Thismethodallowsalsotocoupledifferentdiscretizations,spectralandfiniteelements30,10,andmorerecentlyfiniteelementsandwavelets20.Intheareaofsolutionalgorithms,manyideasthat“work”onstandarddiscretizationshavebeenextendedtothemortarframework(iterativesub-structuring2,1,multigrid24).Themortarelementmethodcanbecastinahybridframework9and,inthiscase,ithasanalternativethatcanbefoundinthemorerecentthreefieldmethod5,25.Inordertoenlargeevenmorethedomainofapplicationofthemortarelementapproach,someattemptshavebeenrecentlyrealizedtoextendittodifferentkindofpartialdifferentialproblemssuchasvariationalinequalitiesforthemodelingofunilateralcontacts.Wereferto14,34whereoptimalconvergenceresultsareproven.Amongthedomainsthatarenotcurrentlycovered,wecanquotetheelectromag-neticwavepropagationfieldandwhatinvolvesthefullsetofMaxwell’sequations.Asfarasweknow,currentlyveryfewefficientsolversexistusingdomaindecompo-sition,andamongthemwereferto7inthe3Dconformingcase,wherethematch-ingconstraintsareexpressedusingLagrangemultipliers41.Themortarapproachseemstobenaturallyfittedtothatcaseandbringaboutalltheadvantagesthathavebeenquotedbefore(towhichcanalsobeaddedtheslidingmeshmethod6,27toapproximateeddycurrentsinturningmachinesorthepotentialinter-processorcommunicationsaving15).AccordingtothelocaldielectricpermittivityεandtoItwastherealsituationatthetime(1997)whenthisworkwasachievedinitsoldversion.Sincethen,somesubstantialadvanceswererealizedonthistopic,werefer,e.g.,to40,26,21.