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1、TR2005-860ADOMAINDECOMPOSITIONDISCRETIZATIONOFPARABOLICPROBLEMSMAKSYMILIANDRYJA∗ANDXUEMINTU†Abstract.Inrecentyears,domaindecompositionmethodshaveattractedmuchattentionduetotheirsuccessfulapplicationtomanyellipticandparabolicproblems.Domaindecompositionmethodstreat
2、problemsbasedonadomainsubstructuring,whichisattractiveforparallelcomputation,duetotheindependenceamongthesubdomains.Inprinciple,domaindecompositionmethodsmaybeappliedtothesystemresultingfromastandarddiscretizationoftheparabolicproblemsor,directly,becarriedoutthrou
3、ghadirectdiscretizationofparabolicproblems.Inthispaper,adirectdomaindecompositionmethodisintroducedtodiscretizetheparabolicproblems.Thestabilityandconvergenceofthisalgorithmareanalyzed,andanO(τ+h)errorboundisprovided.Keywords.domaindecomposition,parabolicproblem,fi
4、niteelementAMSsubjectclassifications.65F10,65N301.Introduction.Domaindecompositionmethodsarebecomingpopularalgo-rithmsforthenumericalsolutionsofpartialdifferentialequations(PDEs)suchasparabolicproblems.Severalstrategiescanbeappliedtoobtainsuchalgorithms.Amongthem,afi
5、rstapproachusesthestandarddiscretizationofparabolicproblems(e.g.,thebackwardEuler,CrankNicolson),followedbyapplyingdomaindecomposi-tionmethodstotheresultingsystems,asaniterativemethodasforellipticproblems(forreferences,see[1],[5]andliteraturetherein).Incontrast,as
6、econdapproachisbasedonthediscretizationoftheparabolicproblemswhichleadstoadomaindecompositionalgorithmasadirectmethod(forreferences,see[3],[6]andtheliter-aturetherein).Thesestrategieshaveproved,theoreticallyandpractically,tobeveryeffectiveforparallelcomputation.Int
7、hispaper,adomaindecompositionmethodisintroducedforparabolicprob-lemsbasedonthesecondapproach.Forasecondorderparabolicequationin(0,T)×Ω,whereΩisapolygonalregionintwodimensionalspace,weconsideranapproxi-mationofaninitial-boundaryvalueproblem.Thisproblemisdirectlydis
8、cretizedbyafinitedifferencemethodwithrespecttothetimevariabletandbyafiniteelementmethodwithrespecttothespatialvariablesx=(x1,x2),leadingtoadirectdomaindeco
