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Convergence of adaptive discontinuous Galerkin methods

Authors: Christian Kreuzer and Emmanuil H. Georgoulis
Journal: Math. Comp. 87 (2018), 2611-2640
MSC (2010): Primary 65N30, 65N12, 65N50, 65N15
Published electronically: February 26, 2018
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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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Additional Information

Christian Kreuzer
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44801 Bochum, Germany

Emmanuil H. Georgoulis
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom; and Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou 157 80, Greece

Keywords: Adaptive discontinuous Galerkin methods, convergence, elliptic problems
Received by editor(s): December 13, 2016
Received by editor(s) in revised form: June 27, 2017
Published electronically: February 26, 2018
Additional Notes: The research of Christian Kreuzer was supported by DFG research grant KR 3984/5-1.
Emmanuil H. Georgoulis acknowledges support by the Leverhulme Trust.
Article copyright: © Copyright 2018 American Mathematical Society

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