Adaptively enriched coarse space for the discontinuous Galerkin multiscale problems

TL;DR

This paper introduces an adaptive coarse space enrichment technique for a domain decomposition preconditioner in discontinuous Galerkin methods, improving efficiency for multiscale elliptic problems with heterogeneous coefficients.

Contribution

It presents a novel adaptive enrichment strategy for the coarse space in a Schwarz preconditioner, ensuring contrast-independent condition numbers.

Findings

Condition number is independent of coefficient contrast with proper enrichment.

Numerical results confirm the effectiveness of the adaptive enrichment.

Preconditioner improves convergence for multiscale problems.

Abstract

In this paper, we propose a two-level overlapping additive Schwarz domain decomposition preconditioner for the symmetric interior penalty discontinuous Galerkin method for the second order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.