Load and memory balanced mesh partitioning for a parallel en(4)

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

G0

Uncoarsening phase3G

GInitial partitioning phase

Fig.3.Theschemaofmultilevelk-waypartitioning.

coarsergraphscreateslevels.Amatchingisoftenusedtocollapse2verticesintoamultivertex.TheSHEMheuristicsformatching,introducedin[5],workswell.

Initialpartitioning.Whenthecoarsestgraphissu cientlysmall,itcanbe

partitionedbyanygraphpartitioningtechnique.

Uncoarsening.AcoarsergraphGlisuncoarsedtoGl 1andthepartitioning

ofGlisprojectedtoGl 1andthenre ned.TheFiduccia-Mattheyses(FM)heuristicsisasimple,fast,andsu cientlygoodoptionforthere nement.Itsearchescandidateverticesinthesetofboundaryvertices,i.e.,verticesadjacenttoavertexinanotherpartition.Thenittriestomoveaselectedvertexintootherpartitions.Themoveisacceptedifoneofthefollowingconditionsisful lled:

1.Thesizeoftheedgecutisdecreasedandthepartitioningremainsbal-anced.

2.Thesizeoftheedgecutisnotincreased,butthebalancingisimproved.Toimprovethebalancingofastronglydisbalancedpartitioning,abalancingstepmaybyadded.ItworksliketheFMheuristics,buttheconditionsforacceptingamovearedi erent:

1.Thebalancingisimproved.

2.Thesizeoftheedgecutisdecreased,butthebalancingisnotworsened.

2.2ReorderingandAssemblingofSubmatrices(Phases2+3)

Afterthedomaindecomposition,theinternalnodesinallpartitionsarereorderedtominimizethesizeoftheenvelope.

De nition3.Inthei-thstepofthefactorizationofasymmetricmatrixA,thewavefrontissetofpairs

<wi(A)={{r,i}: ars=0,r>=i,s=i}.(2)