Overlapping Schwarz methods with adaptive coarse spaces for multiscale problems in 3D

Abstract

We propose two variants of the overlapping additive Schwarz method for the finite element discretization of scalar elliptic problems in 3D with highly heterogeneous coefficients. The methods are efficient and simple to construct using the abstract framework of the additive Schwarz method, and an idea of adaptive coarse spaces. In one variant, the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problems on the faces. In the other variant, it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and subdomain edges. The functions that constitute the coarse spaces are chosen adaptively, and they correspond to the eigenvalues that are smaller than a given threshold. The convergence rate of the preconditioned conjugate gradients method in both cases is shown to be independent of the variations in the coefficients for the sufficient number of eigenfunctions in the coarse space. Numerical results are given to support the theory.