In this paper, we study domain decomposition preconditioners for multiscale flows in high-contrast media. We consider flow equations governed by elliptic equations in heterogeneous media with a large contrast in the coefficients. Our main goal is to develop domain decomposition preconditioners with the condition number that is independent of the contrast when there are variations within coarse regions. This is accomplished by designing coarse-scale spaces and interpolators that represent important features of the solution within each coarse region. The important features are characterized by the connectivities of high-conductivity regions. To detect these connectivities, we introduce an eigenvalue problem that automatically detects high-conductivity regions via a large gap in the spectrum. A main observation is that this eigenvalue problem has a few small, asymptotically vanishing eigenvalues. The number of these small eigenvalues is the same as the number of connected high-conductivity regions. The coarse spaces are constructed such that they span eigenfunctions corresponding to these small eigenvalues. These spaces are used within two-level additive Schwarz preconditioners as well as overlapping methods for the Schur complement to design preconditioners. We show that the condition number of the preconditioned systems is independent of the contrast. More detailed studies are performed for the case when the high-conductivity region is connected within coarse block neighborhoods. Our numerical experiments confirm the theoretical results presented in this paper.

In this paper, robust preconditioners for multiscale flow problems are investigated. We consider elliptic equations with highly varying coefficients. We design and analyze two-level domain decomposition preconditioners that converge independent of the contrast in the media properties. The coarse spaces are constructed using selected eigenvectors of a local spectral problem. Our new construction enriches any given initial coarse space to make it suitable for high-contrast problems. Using the initial coarse space we construct local mass matrices for the local eigenvalue problems. We show that there is a gap in the spectrum of the eigenvalue problem when high-conductivity regions are disconnected. The eigenvectors corresponding to small, asymptotically vanishing eigenvalues are chosen to construct an enrichment of the initial coarse space. Only via a judicious choice of the initial space do we reduce the dimension of the resulting coarse space. Classical coarse basis functions such as multiscale or energy minimizing basis functions can be taken as the basis for the initial coarse space. In particular, if we start with classical multiscale basis, the selected eigenvectors represent only high-conductivity features that cannot be localized within a coarse-grid block, e.g., high-conductivity channels that connect the boundaries of a coarse-grid block. Numerical experiments are presented. The new construction presented here can handle tensor coefficients. The results of this paper substantially extend those presented in [J. Galvis and Y. Efendiev, Multiscale Model. Simul., 8 (2010), pp. 1461–1483], where only scalar coefficients are considered and the coarse space dimension can be very large because the coarse space includes all isolated high-conductivity features that are within a coarse block.

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.