Domain Decomposition Preconditioners for a Discontinuous Galerkin Formulation of a Multiscale Elliptic Problem

TL;DR

This paper introduces domain decomposition preconditioners for a discontinuous Galerkin approach to multiscale elliptic PDEs, achieving coefficient-independent condition number bounds without requiring coefficient continuity across coarse grid boundaries.

Contribution

It develops new domain decomposition preconditioners for DG formulations of multiscale elliptic problems, with proven coefficient-independent bounds and no continuity assumptions.

Findings

Condition number bounds are independent of coefficients.

Both nonoverlapping and overlapping methods are effective.

The analysis extends the theory for DG discretizations of multiscale problems.

Abstract

In this paper, we propose a domain decomposition method for multiscale second order elliptic partial differential equations with highly varying coefficients. The method is based on a discontinuous Galerkin formulation. We present both a nonoverlapping and an overlapping version of the method. We prove that the condition number bound of the preconditioned algebraic system in either case can be made independent of the coefficients under certain assumptions. Also, in our analysis, we do not need to assume that the coefficients are continuous across the coarse grid boundaries. The analysis and the condition number bounds are new, and contribute towards further extension of the theory for the discontinuous Galerkin discretization for multiscale problems.