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 $p$ (${\mathcal{P}}_{p}$-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 ${\mathcal{P}}_{p}$-basis on both polytopic and tensor-product elements with a (standard) DGFEM employing a (mapped) ${\mathcal{Q}}_{p}$-basis. Moreover, a computational example is also presented which demonstrates the performance of the proposed $hp$-version DGFEM on general agglomerated meshes.

DOI: 10.1051/m2an/2015059

Keywords: Discontinuous Galerkin, polygonal elements, polyhedral elements, hp-finite element methods, inverse estimates, ��-basis, PDEs with nonnegative characteristic form

^{1}; Dong, Zhaonan

^{1}; Georgoulis, Emmanuil H.

^{2}; Houston, Paul

^{3}

@article{M2AN_2016__50_3_699_0, author = {Cangiani, Andrea and Dong, Zhaonan and Georgoulis, Emmanuil H. and Houston, Paul}, 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}, volume = {50}, number = {3}, year = {2016}, doi = {10.1051/m2an/2015059}, mrnumber = {3507270}, zbl = {1342.65213}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015059/} }

TY - JOUR AU - Cangiani, Andrea AU - Dong, Zhaonan AU - Georgoulis, Emmanuil H. AU - Houston, Paul TI - $hp$-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 699 EP - 725 VL - 50 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015059/ DO - 10.1051/m2an/2015059 LA - en ID - M2AN_2016__50_3_699_0 ER -

%0 Journal Article %A Cangiani, Andrea %A Dong, Zhaonan %A Georgoulis, Emmanuil H. %A Houston, Paul %T $hp$-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 699-725 %V 50 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015059/ %R 10.1051/m2an/2015059 %G en %F M2AN_2016__50_3_699_0

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 , Volume 50 (2016) no. 3, pp. 699-725. doi : 10.1051/m2an/2015059. http://www.numdam.org/articles/10.1051/m2an/2015059/

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