hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes

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

Abstract

We consider the hp-version interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the advection-diffusion-reaction equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, new hp-version a priori error bounds are derived based on a specific choice of the interior penalty parameter which allows for edge/face-degeneration. The proposed method employs elemental polynomial bases of total degree p (P_p-basis) defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. Numerical experiments highlighting the performance of the proposed DGFEM are presented. In particular, we study the competitiveness of the p-version DGFEM employing a P_p-basis on both polytopic and tensor-product elements with a (standard) DGFEM employing a (mapped) Q_p-basis. Moreover, a computational example is also presented which demonstrates the performance of the proposed hp-version DGFEM on general agglomerated meshes.

Citation

Cangiani, A., Dong, Z., Georgoulis, E. H., & Houston, P. (2016). hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes. ESAIM: Mathematical Modelling and Numerical Analysis, 50(3), 699-725. https://doi.org/10.1051/m2an/2015059

• Journal Article Type: Article
• Acceptance Date: Aug 14, 2015
• Online Publication Date: May 23, 2016
• Publication Date: 2016-05
• Deposit Date: Aug 28, 2015
• Publicly Available Date: May 23, 2016
• Journal: ESAIM: Mathematical Modelling and Numerical Analysis
• Print ISSN: 0764-583X
• Electronic ISSN: 1290-3841
• Publisher: EDP Open
• Peer Reviewed: Peer Reviewed
• Volume: 50
• Issue: 3
• Pages: 699-725
• DOI: https://doi.org/10.1051/m2an/2015059
• Keywords: discontinuous Galerkin; polygonal elements; polyhedral elements; hp-finite element methods; inverse estimates; P-basis; PDEs with nonnegative characteristic form
• Public URL: https://nottingham-repository.worktribe.com/output/788777
• Publisher URL: http://www.esaim-m2an.org/articles/m2an/abs/2016/03/m2an150070/m2an150070.html
• Additional Information: Copyright EDP Sciences, SMAI 2016