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 Galvis and Efendiev
[Multiscale Model. Simul. 8 (2010) 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.