Preconditioning Nonsymmetric and Indefinite Matrices
This chapter describes an approach of preconditioning nonsymmetric and indefinite matrices that can be treated as perturbations of symmetric positive definite ones. Namely, we assume that A = A0 + R where A0 is a s.p.d. matrix and R can be treated as a perturbation of A. More specifically, the main assumption is that in a space complementary to a coarse space of a fixed size, R has a small norm. This is made more precise in what follows.
An additive version of the approach described in what followswas originally considered by Yserentant [Y86]. The Schur complement preconditioner that we present next is found in [V92a]. An equivalent finite element type preconditioner was considered in [Xu92b].
The assumptions made in the present chapter are verified for finite element (nonsymmetric and possibly indefinite) matrices corresponding to general second-order elliptic bilinear forms in Appendix B; see in particular, Theorems B.3 and B.4 for the perturbation approach described in Section 8.2.
KeywordsCoarse Space Interpolation Matrix Coercivity Estimate Present Chapter Coarse Matrix
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