Multilevel Preconditioners for Discontinuous Galerkin Approximations of Elliptic Problems with Jump Coefficients

TL;DR

This paper develops and analyzes multilevel preconditioners for discontinuous Galerkin methods applied to elliptic problems with large coefficient jumps, demonstrating robustness and near-optimality through theoretical analysis and numerical experiments.

Contribution

It introduces a novel splitting approach for IPDG methods that enables the design of robust multilevel preconditioners for problems with jump coefficients.

Findings

Preconditioners are robust against large coefficient jumps.

Preconditioners achieve nearly optimal convergence rates.

Numerical results confirm theoretical robustness and efficiency.

Abstract

We introduce and analyze two-level and multi-level preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents extra difficulties in the analysis which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and nearly-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. The paper includes an Appendix with a collection of proofs of several technical results required for the analysis.