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

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

doi:10.1051/m2an/2015059

h p

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

Cangiani, Andrea ¹ ; Dong, Zhaonan ¹ ; Georgoulis, Emmanuil H.² ; Houston, Paul ³

¹ Department of Mathematics, University of Leicester, Leicester LE1 7RH, 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
³ School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 699-725.

Résumé

We consider the $h p$ -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 $h p$ -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 $h p$ -version DGFEM on general agglomerated meshes.

Reçu le : 2015-04-14

MR Zbl | 3 citations dans Numdam

DOI : 10.1051/m2an/2015059

Classification : 65N30, 65N50, 65N55
Mots-clés : Discontinuous Galerkin, polygonal elements, polyhedral elements, hp-finite element methods, inverse estimates, ��-basis, PDEs with nonnegative characteristic form

Affiliations des auteurs :

Cangiani, Andrea ¹ ; Dong, Zhaonan ¹ ; Georgoulis, Emmanuil H. ² ; Houston, Paul ³

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     title = {$hp${-Version} discontinuous {Galerkin} methods for advection-diffusion-reaction problems on polytopic meshes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {699--725},
     publisher = {EDP-Sciences},
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Cangiani, Andrea; Dong, Zhaonan; Georgoulis, Emmanuil H.; Houston, Paul. $hp$-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 699-725. doi : 10.1051/m2an/2015059. http://archive.numdam.org/articles/10.1051/m2an/2015059/

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