Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes

TL;DR

This paper develops robust multilevel preconditioners for spectral Discontinuous Galerkin methods that maintain bounded condition numbers despite large local polynomial degree variations, improving solver efficiency.

Contribution

It introduces a novel auxiliary space preconditioning framework using Legendre-Gauss-Lobatto grids and wavelet techniques for elliptic problems with variable polynomial degrees.

Findings

Preconditioners achieve uniform boundedness of condition numbers under mild grading constraints.

The modular preconditioner design allows integration with domain decomposition methods.

Experimental results confirm the theoretical robustness and efficiency of the proposed preconditioners.

Abstract

This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of the condition numbers that is logarithmic in the polynomial degrees when all degrees are equal and quadratic otherwise, our main objective is to realize full robustness with respect to arbitrarily large locally varying polynomial degrees degrees, i.e., under mild grading constraints condition numbers stay uniformly bounded with respect to the mesh size and variable degrees. The conceptual foundation of the envisaged preconditioners is the auxiliary space method. The main conceptual ingredients that will be shown in this framework to yield "optimal" preconditioners in the above sense are Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic nested dyadic grids as well as specially adapted wavelet preconditioners for the resulting low order auxiliary problems. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations. One of the components can, for instance, be conveniently combined with domain decomposition, at the expense though of a logarithmic growth of condition numbers. Our analysis is complemented by quantitative experimental studies of the main components.