Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods

TL;DR

This paper proves the convergence of weakly penalized adaptive discontinuous Galerkin methods, establishing a contraction property under minimal stabilization assumptions through an auxiliary solution approach.

Contribution

It introduces a convergence proof for adaptive DG methods with weak penalization, relying only on sufficiently large stabilizing parameters, unlike previous works.

Findings

Convergence of weakly penalized adaptive DG methods is established.

A contraction property is derived under minimal stabilization assumptions.

An auxiliary solution construction facilitates the convergence analysis.

Abstract

In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by a simple post processing. Based on the auxiliary solution, we define the adaptive algorithm which guides to the convergence of adaptive discontinuous Galerkin methods.