TY - JOUR

T1 - Combinatorial preconditioners for scalar elliptic finite-element problems

AU - Avron, Haim

AU - Chen, Doron

AU - Shklarski, Gil

AU - Toledo, Sivan

PY - 2009

Y1 - 2009

N2 - We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDE's). The solver splits the collection {Ke} of element matrices into a subset of matrices that are approximable by diagonally dominant matrices and a subset of matrices that are not approximable. The approximable K e's are approximated by diagonally dominant matrices Le's that are assembled to form a global diagonally dominant matrix L. A combinatorial graph algorithm then approximates L by another diagonally dominant matrix M that is easier to factor. Finally, M is added to the inapproximable elements to form the preconditioner, which is then factored. When all the element matrices are approximable, which is often the case, the preconditioner is provably efficient. Approximating element matrices by diagonally dominant ones is not a new idea, but we present a new approximation method which is both efficient and provably good. The splitting idea is simple and natural in the context of combinatorial preconditioners, but hard to exploit in other preconditioning pa radigms. Experimental results show that on problems in which some of the Ke's are ill conditioned, our new preconditioner is more effective than an algebraic multigrid solver, than an incomplete-fa ctorization preconditioner, and than a direct solver.

AB - We present a new preconditioner for linear systems arising from finite-element discretizations of scalar elliptic partial differential equations (PDE's). The solver splits the collection {Ke} of element matrices into a subset of matrices that are approximable by diagonally dominant matrices and a subset of matrices that are not approximable. The approximable K e's are approximated by diagonally dominant matrices Le's that are assembled to form a global diagonally dominant matrix L. A combinatorial graph algorithm then approximates L by another diagonally dominant matrix M that is easier to factor. Finally, M is added to the inapproximable elements to form the preconditioner, which is then factored. When all the element matrices are approximable, which is often the case, the preconditioner is provably efficient. Approximating element matrices by diagonally dominant ones is not a new idea, but we present a new approximation method which is both efficient and provably good. The splitting idea is simple and natural in the context of combinatorial preconditioners, but hard to exploit in other preconditioning pa radigms. Experimental results show that on problems in which some of the Ke's are ill conditioned, our new preconditioner is more effective than an algebraic multigrid solver, than an incomplete-fa ctorization preconditioner, and than a direct solver.

KW - Combinatorial preconditioners

KW - Finite elements

KW - Preconditioning

KW - Support preconditioners

UR - http://www.scopus.com/inward/record.url?scp=72449196960&partnerID=8YFLogxK

U2 - 10.1137/060675940

DO - 10.1137/060675940

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AN - SCOPUS:72449196960

SN - 0895-4798

VL - 31

SP - 694

EP - 720

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 2

ER -