A uniform additive Schwarz preconditioner for the $hp$-version of Discontinuous Galerkin approximations of elliptic problems

TL;DR

This paper introduces a uniform additive Schwarz preconditioner for high-order Discontinuous Galerkin methods applied to elliptic problems, ensuring spectral bounds are independent of discretization parameters.

Contribution

It presents a novel preconditioner based on space splitting that maintains spectral bounds uniformly across mesh size, polynomial degree, and penalization coefficient.

Findings

Spectral bounds are uniform with respect to discretization parameters.

The preconditioner performs well in numerical simulations.

The approach effectively handles nearly-singular problems.

Abstract

In this paper we design and analyze a uniform preconditioner for a class of high order Discontinuous Galerkin schemes. The preconditioner is based on a space splitting involving the high order conforming subspace and results from the interpretation of the problem as a nearly-singular problem. We show that the proposed preconditioner exhibits spectral bounds that are uniform with respect to the discretization parameters, i.e., the mesh size, the polynomial degree and the penalization coefficient. The theoretical estimates obtained are supported by several numerical simulations.