Robust Preconditioners for DG-Discretizations with Arbitrary Polynomial Degrees

TL;DR

This paper introduces a multi-stage preconditioner for DG discretizations of elliptic problems that ensures uniformly bounded condition numbers, supporting flexible polynomial degrees and mesh grading.

Contribution

It develops a novel multi-stage preconditioner based on auxiliary space methods and spectral element techniques, enhancing the efficiency of solving DG discretizations with arbitrary polynomial degrees.

Findings

Preconditioner achieves uniformly bounded condition numbers.

Numerical studies illustrate the effectiveness of the preconditioner.

Supports full flexibility of DG methods under mild grading conditions.

Abstract

Discontinuous Galerkin (DG) methods offer an enormous flexibility regarding local grid refinement and variation of polynomial degrees for a variety of different problem classes. With a focus on diffusion problems, we consider DG discretizations for elliptic boundary value problems, in particular the efficient solution of the linear systems of equations that arise from the Symmetric Interior Penalty DG method. We announce a multi-stage preconditioner which produces uniformly bounded condition numbers and aims at supporting the full flexibility of DG methods under mild grading conditions. The constructions and proofs are detailed in an upcoming series of papers by the authors. Our preconditioner is based on the concept of the auxiliary space method and techniques from spectral element methods such as Legendre-Gau{\ss}-Lobatto grids. The presentation for the case of geometrically conforming meshes is complemented by numerical studies that shed some light on constants arising in four basic estimates used in the second stage.