Schur complement preconditioners for anisotropic problems

Abstract

We present two new variants of Schur complement domain decomposition preconditioners suitable for 2D anisotropic problems. These variants are based on adaptations of the probing idea, described by Chan et al (1992 Fifth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, Philadelphia: SIAM, pp 236-249), used in conjunction with a coarse grid approximation as introduced by Bramble et al (1986 Math. Comput. 47, 103-134). The new methods are specifically designed for situations where the coupling between neighbouring interfaces is stronger than the coupling within an interface. Taking into account this strong coupling, one variant uses a multicolour probing technique to avoid distortion in the probe approximations that appear when using the method proposed by Chan et al. The second technique uses additional probe matrices to approximate not only the coupling within the interfaces but also the coupling between interface points across the subdomains. This latter procedure looks somewhat like an alternating line relaxation procedure and was motivated by the success of line relaxation within the multigrid method for anisotropic problems, see Brandt (1977 Math. Comput. 31, 333-390). To assess the relevance of the new preconditioners, we compare their numerical behaviour with well-known robust preconditioners such as the balanced Neumann-Neumann method proposed by Mandel (1993 Commun. Numer, Methods Eng. 9, 233-241).