Load and memory balanced mesh partitioning for a parallel en(2)

Abstract. We use a parallel direct solver based on the Schur complement method for solving large sparse linear systems arising from the finite element method. A domain decomposition of a problem is performed using a graph partitioning. It results in sparse

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132610beh143711cfi154816(a)(b)(c)

Fig.1.Aquadrilateralmesh(a)with9elementsa–iand16nodes1–16.Thedual(b)andthenodal(c)graphderivedfromthemesh.

6.Backsubstitutiononsubdomains.

Thedomaindecomposition,andoptionallyorderingofnodes,aredoneasprepro-cessingsteps.Thesolverdoestherest.Phase3isthemostmemoryconsumingandPhase4isthemostcomputationallyintensivepartofthewholesolution.

Multileveltoolsarewidelyusedtosolvetheproblemofdomaindecompo-sition.Thedualgraphispartitionedintokparts,inducingsubdomainsandcorrespondingsubmatrices,sothatthesizesofsubmatricesareroughlyequal.De nition1.ConsideragraphG=(V,E)andanintegerk>=2.Anedgecut(nodecut)isasetofedges(vertices,respectively)whoseremovaldividesthegraphintoatleastkpartitions.Thek-waygraphpartitioningproblemisto.partitionVintokpairwisedisjointsubsetsV1,V2,...,Vksuchthat|Vi|=|V|/kandthesizeoftheedgecutisminimized.Apartitioningofagraphbyanodecutissimilar.

Eventhoughthesizesofsubmatricesareroughlyequal,theirmemoryre-quirementsortheirfactorizationtimeinPhase4arenotequal.Tode nethisformally,weusethetermqualitytodenotethememoryorcomputationalcom-plexity.

De nition2.Givenanunbalancingthresholdδ>=1,wesaythatapartitioningV1,V2,...,Vkwithasetofqualities{q1,q2,...,qk}isbalancedif

δ>qi)k/=(maxi=1

ApartitionViisoverbalancedifδ<qik/

ifatleastonepartitionisoverbalanced.kk i=1iqi.qi.(1) Thepartitioningisdisbalanced

Ingeneral,thequalitiesofsubmatricesarein uencedbytheorderingofvariables,aswasalreadymentionedin[2,3].

Inthispaper,wedescribeanovelapproachtothedomaindecompositionthatresultsintopartitioningwithbalancedmemoryrequirementsorbalanced