Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms

TL;DR

This paper develops a robust abstract framework for domain decomposition preconditioners that ensure stable and efficient iterative solutions for symmetric positive definite problems, including complex porous media equations.

Contribution

It introduces a new abstract framework for stable space decompositions applicable to various PDEs, with coarse spaces built from eigenfunctions, ensuring robustness against mesh and contrast variations.

Findings

Condition numbers are uniformly bounded with respect to contrast and mesh parameters.

The framework applies to scalar elliptic, Stokes, and Brinkman's equations.

Numerical experiments confirm the theoretical robustness and efficiency.

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 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.