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

France Adnene

Description

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

Sujets

• Sabatier

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 17
• Langue: English

Extrait

1 Wedescribethe mortarelementmethod forthetwo-dimensionaledgefiniteelementsof class H (curl).Thesefiniteelementsarecurrentlyusedforthediscretizationofvarious modelscomingfromtheMaxwellequationsandusingtheminamortarframeworkhas severalinterestingapplicationsintheelectromagneticandelectrotechnicaldomains.We developsometechnicaltoolsnecessarytoperformthenumericalanalysisofthisapproach. Then,weprovesomeoptimalapproximationresultsandweillustratethetheorybysome numericalexperiences. Keywords :edgefiniteelements,non-matchinggrids,mortarelementmethod 1.Introduction Domaindecompositiontechniqueshavebecomeastandardwaytoincreasethesize oftheproblemsthatcanbetreatedthroughnumericalsimulation.Thesetechniques ∗ Theauthorsrecallthat,despiteitslatepublication,thispaperisthestartingsteptowardthe applicationofthemortarelementmethodtoelectromagnetism. Mortaringthetwo-dimensionaledgefiniteelementsfor thediscretizationofsomeelectromagneticmodels ∗ FrancescaRAPETTI LaboratoireJ.-A.Dieudonne´,UMR6621CNRS,Univ.NiceetSophia-Antipolis, ParcValrose,06108Nicecedex02,France. [e-mail] Adne`neBENABDALLAH ApplicationsScientifiquespourleCalculIntensif,UPR9069CNRS, Baˆt.506,Universite´ParisSud,91405Orsay,France. [e-mail] YvonMADAY LaboratoireJ.-L.Lions,UMR7583CNRS,Univ.PierreetMarieCurie, 4placeJussieu,75252Pariscedex05,France. [e-mail] FakerBENBELGACEM Mathe´matiquespourl’IndustrieetlaPhysique,UMR5640CNRS,Univ.PaulSabatier, 118routedeNarbonne,31062ToulouseCedex04,France. [e-mail] ynapmoCgnihsilbuPcfiitneicSdlroWc ❢ secneicSdeilppAnisdohteMdnasledoMlacitamehtaM

2A.BenAbdallah,F.BenBelgacem,Y.Maday,F.Rapetti allowtouseefficientlytheparallelpotentialityofcomputersandtodesignrobust iterativesolverstoacceleratetheaccesstothenumerical(approximated)solution. Whendealingwithellipticorparabolicproblems,inordertoincreaseevenmore theflexibilityofthedomaindecompositionapproach,themortarelementmethod 19,10 allowstouse,ineachsub-domainofthedecomposition,theproperdiscretiza- tion(offiniteelement,spectral,finitevolumeorofwavelettype)wellsuitedtothe local(relatedtothesub-domain)solutionbehaviorandgeometricalfeatures.The resultingapproximationisoptimalintermsoferrorestimatesandconvergencerates sincethelocaldiscretizationparameterscanbechosenindependentlyovereachsub- domaininsuchawaythateachlocalcontributiontotheerrorisbalanced.This discretizationparametercaneitherbedesigned apriori or aposteriori 42,18 ina suitableway. Themortarelementmethodhasreceivedinthelastdecadealotofattention anditiscurrentlyusedinmanyareaofthenumericalsimulationsuchas,forex- ample,forthespectralapproximationofNavierStokesequations(intheprimal variablesformulation 6 , 12 ,andinthe ψ or( ψ,ω )formulation 8 ),forthefiniteele- mentapproximationofNavierStokesequations(intheprimalvariablesformulation 3 , 11 ),forthefiniteelementapproachofelasticity 37 , 36 .Thismethodallowsalsoto coupledifferentdiscretizations,spectralandfiniteelements 30,10 ,andmorerecently finiteelementsandwavelets 20 .Intheareaofsolutionalgorithms,manyideasthat “work”onstandarddiscretizationshavebeenextendedtothemortarframework (iterativesub-structuring 2,1 ,multigrid 24 ). Themortarelementmethodcanbecastinahybridframework 9 and,inthis case,ithasanalternativethatcanbefoundinthemorerecentthreefieldmethod 5 , 25 .Inordertoenlargeevenmorethedomainofapplicationofthemortarelement approach,someattemptshavebeenrecentlyrealizedtoextendittodifferentkind ofpartialdifferentialproblemssuchasvariationalinequalitiesforthemodelingof unilateralcontacts.Wereferto 14 , 34 whereoptimalconvergenceresultsareproven. Amongthedomainsthatarenotcurrentlycovered, † wecanquotetheelectromag- neticwavepropagationfieldandwhatinvolvesthefullsetofMaxwell’sequations. Asfarasweknow,currentlyveryfewefficientsolversexistusingdomaindecompo- sition,andamongthemwereferto 7 inthe3Dconformingcase,wherethematch- ingconstraintsareexpressedusingLagrangemultipliers 41 .Themortarapproach seemstobenaturallyfittedtothatcaseandbringaboutalltheadvantagesthat havebeenquotedbefore(towhichcanalsobeaddedtheslidingmeshmethod 6 , 27 toapproximateeddycurrentsinturningmachinesorthepotentialinter-processor communicationsaving 15 ).Accordingtothelocaldielectricpermittivityεandto † Itwastherealsituationatthetime(1997)whenthisworkwasachievedinitsoldversion.Since then,somesubstantialadvanceswererealizedonthistopic,werefer,e.g.,to 40 , 26 , 21 .