hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes

Cangiani, Andrea; Georgoulis, Emmanuil H.; Houston, Paul

Abstract

An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analysed. Utilizing a bounding box concept, the method employs elemental polynomial bases of total degree p (P_p-basis) defined on the physical space, without the need to map from a given reference or canonical frame. This, together with a new specific choice of the interior penalty parameter which allows for face-degeneration, ensures that optimal a priori bounds may be established, for general meshes including polygonal elements with degenerating edges in two dimensions and polyhedral elements with degenerating faces and/or edges in three dimensions. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the p-version DGFEM employing a P_p-basis in comparison to the conforming p-version finite element method on tensor-product elements is studied numerically for a simple test problem.

Citation

Cangiani, A., Georgoulis, E. H., & Houston, P. (2014). hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Mathematical Models and Methods in Applied Sciences, 24(10), 2009-2041. https://doi.org/10.1142/S0218202514500146

• Journal Article Type: Article
• Acceptance Date: Dec 10, 2013
• Online Publication Date: Mar 11, 2014
• Publication Date: 2014-09
• Deposit Date: Aug 26, 2015
• Publicly Available Date: Aug 26, 2015
• Journal: Mathematical Models and Methods in Applied Sciences
• Print ISSN: 0218-2025
• Electronic ISSN: 0218-2025
• Publisher: World Scientific
• Peer Reviewed: Peer Reviewed
• Volume: 24
• Issue: 10
• Pages: 2009-2041
• DOI: https://doi.org/10.1142/S0218202514500146
• Keywords: Discontinuous Galerkin, polygonal elements, polyhedral elements, hp-finite elements, inverse estimates, P-basis
• Public URL: https://nottingham-repository.worktribe.com/output/725243
• Publisher URL: http://www.worldscientific.com/doi/abs/10.1142/S0218202514500146
• Additional Information: Electronic version of an article published in Mathematical Models and Methods, vol. 24, issue 10, 2014, p. 2009-2041, doi: 10.1142/S0218202514500146 ©World Scientific Publishing Company http://www.worldscientific.com/worldscinet/m3as