  An abstract framework for constructing stable decompositions of the spaces
corresponding to general symmetric positive definite problems into "local"
subspaces and a global "coarse" space is developed. Particular applications of
this abstract framework include practically important problems in porous media
applications such as: the scalar elliptic (pressure) equation and the stream
function formulation of its mixed form, Stokes' and Brinkman's equations. The
constant in the corresponding abstract energy estimate is shown to be robust
with respect to mesh parameters as well as the contrast, which is defined as
the ratio of high and low values of the conductivity (or permeability). The
derived stable decomposition allows to construct additive overlapping Schwarz
iterative methods with condition numbers uniformly bounded with respect to the
contrast and mesh parameters. The coarse spaces are obtained by patching
together the eigenfunctions corresponding to the smallest eigenvalues of
certain local problems. A detailed analysis of the abstract setting is
provided. The proposed decomposition builds on a method of Efendiev and Galvis,
Multiscale Model. Simul., 8 (2010), pp. 1461--1483, developed for second order
scalar elliptic problems with high contrast. Applications to the finite element
discretizations of the second order elliptic problem in Galerkin and mixed
formulation, the Stokes equations, and Brinkman's problem are presented. A
number of numerical experiments for these problems in two spatial dimensions
are provided.
