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Modified Block SSOR Preconditioners for Symmetric Positive Definite Linear Systems

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  • Zhong-Zhi Bai

Abstract

A class of modified block SSOR preconditioners is presented for the symmetric positive definite systems of linear equations, whose coefficient matrices come from the hierarchical-basis finite-element discretizations of the second-order self-adjoint elliptic boundary value problems. These preconditioners include a block SSOR iteration preconditioner, and two inexact block SSOR iteration preconditioners whose diagonal matrices except for the (1,1)-block are approximated by either point symmetric Gauss–Seidel iterations or incomplete Cholesky factorizations, respectively. The optimal relaxation factors involved in these preconditioners and the corresponding optimal condition numbers are estimated in details through two different approaches used by Bank, Dupont and Yserentant (Numer. Math. 52 (1988) 427–458) and Axelsson (Iterative Solution Methods (Cambridge University Press, 1994)). Theoretical analyses show that these modified block SSOR preconditioners are very robust, have nearly optimal convergence rates, and especially, are well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes. Copyright Kluwer Academic Publishers 2001

Suggested Citation

  • Zhong-Zhi Bai, 2001. "Modified Block SSOR Preconditioners for Symmetric Positive Definite Linear Systems," Annals of Operations Research, Springer, vol. 103(1), pages 263-282, March.
    • Handle: RePEc:spr:annopr:v:103:y:2001:i:1:p:263-282:10.1023/a:1012915424955
    DOI: 10.1023/A:1012915424955
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    Cited by:

    1. Dai, Ping-Fan & Li, Jicheng & Bai, Jianchao & Qiu, Jinming, 2019. "A preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 542-551.
    2. Fan, Hong-tao & Zhu, Xin-yun, 2015. "A generalized relaxed positive-definite and skew-Hermitian splitting preconditioner for non-Hermitian saddle point problems," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 36-48.

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