Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Abstract

Abstract. 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.Mathematics Subject Classification. 65F10, 65N20, 65N22, 65N30, 65N55.

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

“…In many cases, this is not feasible and so recently, for scalar elliptic problems, these results have been extended to operator dependent coarse spaces and to coefficients that are not resolved by the subdomain partition, see e.g. [7,8,12,[14][15][16]24,27,28], as well as [32] and the references therein.…”

Additive Schwarz preconditioner for the finite volume element discretization of symmetric elliptic problems

“…[18,19,27], or in extreme cases to enrich it with eigenfunctions corresponding to bad eigen modes of some local eigenvalue problems, cf. [14,21,31]. These are subjects for future investigation.…”

“…[18,19,27], or by enriching the coarse space with eigenfunctions from local eigenvalue problems, cf. [10,14,21,31], it will be possible to extend the methods to effectively deal with highly heterogeneous coefficients.…”

Robust Coarsening in Multiscale PDEs

“…Several approaches have been studied in [2,5,6,7,8,9,17,18,19] to provide constants in (3) that are independent of α (or at least of the contrast in α) for various coarse spaces. However, in most cases the constants are not independent of H ε , where ε is the minimal length scale at which α varies in the regions where ∇χ k = 0.…”

“…We review some recent papers on the topic mainly by the author (jointly with co-workers), as well as by Efendiev et al All the results apply immediately also to multiplicative, hybrid and non-overlapping versions of the Schwarz method (see [9,18] for some explicit comments). Many of the results can be extended to a multilevel theory [18,5].…”