Convergence of adaptive discontinuous Galerkin methods (corrected version of [Math. Comp. 87 (2018), no. 314, 2611--2640])

TL;DR

This paper establishes a comprehensive convergence theory for adaptive discontinuous Galerkin methods applied to elliptic PDEs, encompassing popular schemes and marking strategies, with analysis based on a novel limit space interpolation.

Contribution

It introduces a general convergence framework for adaptive DG methods that applies to various schemes and marking strategies, using a new limit space interpolation approach.

Findings

Convergence proven for SIPG, NIPG, LDG schemes.

Applicable with minimal penalty parameter requirements.

Extends convergence results from conforming to non-conforming methods.

Abstract

We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the presented result is, that it holds for penalty parameters only necessary for the standard analysis of the respective scheme. The analysis is based on a quasi interpolation into a newly developed limit space of the adaptively created non-conforming discrete spaces, which enables to generalise the basic convergence result for conforming adaptive finite element methods by Morin, Siebert, and Veeser [A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 2008, 18(5), 707--737].