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Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. (English) Zbl 1272.65098
Authors’ 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 J. Galvis and Y. Efendiev [Multiscale Model. Simul. 8, No. 4, 1461–1483 (2010; Zbl 1206.76042)] 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 is provided.

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
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