Convergence of adaptive discontinuous Galerkin methods

Kreuzer, Christian; Georgoulis, Emmanuil

doi:10.1090/mcom/3318

Convergence of adaptive discontinuous Galerkin methods
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by Christian Kreuzer and Emmanuil H. Georgoulis;

Math. Comp. 87 (2018), 2611-2640

DOI: https://doi.org/10.1090/mcom/3318

Published electronically: February 26, 2018

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Corrigendum: Math. Comp. 90 (2021), 637-640.

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 us 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].

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