Geodesic
hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes
Cangiani, Andrea1  ; Dong, Zhaonan1 ; Georgoulis, Emmanuil H.2  ; Houston, Paul3
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

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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.

Reçu le :
DOI : 10.1051/m2an/2015059
Classification : 65N30, 65N50, 65N55
Keywords: Discontinuous Galerkin, polygonal elements, polyhedral elements, hp-finite element methods, inverse estimates, ��-basis, PDEs with nonnegative characteristic form

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 
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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 },
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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

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