Combinatorial preconditioners for scalar elliptic finite-element problems

Haim Avron, Doron Chen, Gil Shklarski, Sivan Toledo

Research output: Contribution to journalArticlepeer-review

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 {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.

Original languageEnglish
Pages (from-to)694-720
Number of pages27
JournalSIAM Journal on Matrix Analysis and Applications
Volume31
Issue number2
DOIs
StatePublished - 2009

Keywords

  • Combinatorial preconditioners
  • Finite elements
  • Preconditioning
  • Support preconditioners

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