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

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

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 3, pp. 699-725.

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.

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

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