Load and memory balanced mesh partitioning for a parallel en(5)

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

TheenvelopeofAisthen(symbol\isthesetdi erence)

Env(A)=n

i=1wi(A)\{i,i}.(3)

TheenvelopemethodstoresthepartsofAinsideenvelopes,whichrepresentsthemainportionofthememoryconsumptionofthesolver.TheSIFELsolverusestheSloanreorderingalgorithm[9],butotheralgorithms(RCM,hybridalgorithm)canbeusedaswell.

Everynodeigeneratesdiequations,wherediisthenumberofDOFs.Letniandnbdenotethenumberofinternalandboundary,respectively,variables.Clearly,ni+nb=n.TheSIFELsolverreadsthesubdomainsandassemblessubmatriceswithnodesinagivenorder.Theboundarynodescomelast.Wees-timatememoryrequirementsW(A)oftheenvelopesolveranditscomputationalcomplexityOP(A)by

W(A)=

OP(A)=n i=1ni

i=1|wi(A)|=|Env(A)|+n|wi(A)|2.(4)(5)

3TheNewRe nementHeuristicswithQualityBalancingInthissection,weexplainthemainideasofloadandmemorybalancingfortheparallelenvelopemethod.Atthebeginning,itmustbedecidedwhichqualitywillbebalanced:thecomputationalloadorthememorycomplexity.Thenthepartitioningisstarted.Toestimatethequality,thepartitionermusthavetheinformationaboutthestructureofthematrixA.SincethedualgraphGDdoesnotsu ce,thepartitionermusthaveatdisposalalsotheweightednodalgraphGN,inwhicheachvertexislabelledwiththenumberofDOFsofthenode.WepartitionGDandprojectthepartitioningofGDtoGNtoestimatethequality.So,thepartitionerneedsalsotheinformationabouttherelationbetweenGDandGN.

Weuseamultilevelk-waygraphpartitioningasdescribedinSect.2.1.The rsttwophasesareperformedunchangedasinMETIS.TheSHEMheuristicsisusedforthecoarseningandthemultilevelgraphbisectionisusedfortheinitialpartitioning.Wehavemodi edtheconditionsofmoveacceptanceofthere nementheuristicsintheuncoarseningphaseandextendeditwithaqualityestimationprocess.ThisnewheuristicsiscalledQB.Assumethatthecurrentlevelisl.SimilartoFM,QBchoosesavertexasacandidateformovingfromaDsourcepartitionGDl,stoatargetpartitionGl,t.Todecidewhetherthemovewill

Dbeaccepted,QBstartsforbothpartitionsGDl,sandGl,ttheestimationprocess,

sketchedonFig.4,toobtaininformationaboutbalancingofthequalities.DFirst,partitionGDl,p,p∈{s,t},isprojectedtoG0,p,theoriginalpartition.(Thisisskippedinthe nallevel,wherel=0).Afterthat,allnodesbelonging