Schur complement preconditioners for anisotropic problems

We present two new variants of Schur complement domain decomposition precon ditioners 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, P hiladelphia: SIAM, pp 236-249), used in conjunction with a coarse grid appr oximation 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 in terface. Taking into account this strong coupling, one variant uses a multi colour probing technique to avoid distortion in the probe approximations th at appear when using the method proposed by Chan et al. The second techniqu e uses additional probe matrices to approximate not only the coupling withi n the interfaces but also the coupling between interface points across the subdomains. This latter procedure looks somewhat like an alternating line r elaxation procedure and was motivated by the success of line relaxation wit hin the multigrid method for anisotropic problems, see Brandt (1977 Math. C omput. 31, 333-390). To assess the relevance of the new preconditioners, we compare their numerical behaviour with well known robust preconditioners s uch as the balanced Neumann-Neumann method proposed by Mandel (1993 Commun. Numer Methods Eng. 9, 233-241).