Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems
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Avron-2009-COMBINATORIAL PRECON.pdf
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1.14 MB
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Author(s)
Avron, Haim
Chen, Doron
Shklarski, Gil
Toledo, Sivan
Date Issued
June 2009
Journal
SIAM Journal on Matrix Analysis and Applications
Publisher
Society for Industrial and Applied Mathematics
Citation
Avron, Haim et al. “Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems.” SIAM Journal on Matrix Analysis and Applications 31.2 (2009): 694-720. ©2009 Society for Industrial and Applied Mathematics
Version
Final published version
Abstract
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 $\{K_{e}\}$ 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 $L_{e}$'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 paradigms. Experimental results show that on problems in which some of the $K_{e}$'s are ill conditioned, our new preconditioner is more effective than an algebraic multigrid solver, than an incomplete-factorization preconditioner, and than a direct solver.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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DOI of Published Version
http://dx.doi.org/10.1137/060675940
