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Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

Kawecki, EL; (2020) Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods. Numerical Methods for Partial Differential Equations 10.1002/num.22489. (In press). Green open access

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Abstract

In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs’ inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp-DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.

Type: Article
Title: Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods
Open access status: An open access version is available from UCL Discovery
DOI: 10.1002/num.22489
Publisher version: https://doi.org/10.1002/num.22489
Language: English
Additional information: This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. http://creativecommons.org/licenses/by/4.0/
Keywords: discontinuous Galerkin, finite element method, numerical analysis, partial differential equations
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: https://discovery.ucl.ac.uk/id/eprint/10102461
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