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

¹ Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
Andrea.Cangiani@le.ac.uk; zd14@le.ac.uk
² Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK & School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780 Athens, Greece
Emmanuil.Georgoulis@le.ac.uk
³ School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
Paul.Houston@nottingham.ac.uk

Received: 14 April 2015
Revised: 10 August 2015

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.