Nonoverlapping spectral additive Schwarz methods for hybrid discontinuous Galerkin discretizations

Abstract

Abstract This paper develops exact, inexact and some new variants of nonoverlapping spectral additive Schwarz (NOSAS) methods for hybrid discontinuous Galerkin discretizations. The NOSAS are nonoverlapping preconditioners designed for elliptic problems with highly heterogeneous coefficients, and the subdomain interactions are via the interface problem. The interface problem has two components, a local one with degrees of freedom on the interface of the subdomains and a global one based on a few generalized eigenvectors. These generalized eigenvectors are selected to guarantee the robustness of the NOSAS with respect to heterogeneous diffusion coefficients and the number of subdomains. A unique feature of NOSAS is that the corresponding generalized eigenvalue problems are obtained from the Neumann matrix of the nonoverlapping subdomain separately. Additionally, the number of these few generalized eigenvectors can be characterized by the geometry of the coefficients for the case of two coefficients with high contrast. We prove that the condition number of the NOSAS does not depend on the coefficients. With a specific choice of the threshold for selecting these few eigenvalues, we can recover similar convergence rates as we would have obtained for constant coefficients.