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An abstract framework for constructing stable decompositions of the spacescorresponding to general symmetric positive definite problems into "local"subspaces and a global "coarse" space is developed. Particular applications ofthis abstract framework include practically important problems in porous mediaapplications such as: the scalar elliptic (pressure) equation and the streamfunction formulation of its mixed form, Stokes' and Brinkman's equations. Theconstant in the corresponding abstract energy estimate is shown to be robustwith respect to mesh parameters as well as the contrast, which is defined asthe ratio of high and low values of the conductivity (or permeability). Thederived stable decomposition allows to construct additive overlapping Schwarziterative methods with condition numbers uniformly bounded with respect to thecontrast and mesh parameters. The coarse spaces are obtained by patchingtogether the eigenfunctions corresponding to the smallest eigenvalues ofcertain local problems. A detailed analysis of the abstract setting isprovided. The proposed decomposition builds on a method of Efendiev and Galvis,Multiscale Model. Simul., 8 (2010), pp. 1461--1483, developed for second orderscalar elliptic problems with high contrast. Applications to the finite elementdiscretizations of the second order elliptic problem in Galerkin and mixedformulation, the Stokes equations, and Brinkman's problem are presented. Anumber of numerical experiments for these problems in two spatial dimensionsare provided.

Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms