An abstract framework for constructing stable decompositions of the spaces correspondingto general symmetric positive definite problems into “local” subspaces and a global“coarse” space is developed. Particular applications of this abstract framework includepractically important problems in porous media applications such as: the scalar elliptic(pressure) equation and the stream function formulation of its mixed form, Stokes’ andBrinkman’s equations. The constant in the corresponding abstract energy estimate is shownto be robust with respect to mesh parameters as well as the contrast, which is defined asthe ratio of high and low values of the conductivity (or permeability). The derived stabledecomposition allows to construct additive overlapping Schwarz iterative methods withcondition numbers uniformly bounded with respect to the contrast and mesh parameters. Thecoarse spaces are obtained by patching together the eigenfunctions corresponding to thesmallest eigenvalues of certain local problems. A detailed analysis of the abstractsetting is provided. The proposed decomposition builds on a method of Galvis and Efendiev[Multiscale Model. Simul. 8 (2010) 1461–1483] developedfor second order scalar elliptic problems with high contrast. Applications to the finiteelement discretizations of the second order elliptic problem in Galerkin and mixedformulation, the Stokes equations, and Brinkman’s problem are presented. A number ofnumerical experiments for these problems in two spatial dimensions are provided.