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$hp$-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes

Cangiani, Andrea, Dong, Zhaonan and Georgoulis, Emmanuil H. 2017. $hp$-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes. SIAM Journal on Scientific Computing 39 (4) , A1251-A1279. 10.1137/16M1073285

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We present a new hp-Version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise to prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of total degree, say $p$, defined in the physical coordinate system, as opposed to standard dG time-stepping methods whereby spatial elemental bases are tensorized with temporal basis functions. This approach leads to a fully discrete hp-dG scheme using fewer degrees of freedom for each time step, compared to dG time-stepping schemes employing a tensorized space-time basis, with acceptable deterioration of the approximation properties. A second key feature of the new space-time dG method is the incorporation of very general spatial meshes consisting of possibly polygonal/polyhedral elements with an arbitrary number of faces. A priori error bounds are shown for the proposed method in various norms. An extensive comparison among the new space-time dG method, the (standard) tensorized space-time dG methods, the classical dG time-stepping, and the conforming finite element method in space is presented in a series of numerical experiments.

Item Type: Article
Date Type: Published Online
Status: Published
Schools: Mathematics
Publisher: Society for Industrial and Applied Mathematics
ISSN: 1064-8275
Date of First Compliant Deposit: 16 December 2019
Date of Acceptance: 27 April 2017
Last Modified: 24 Nov 2020 16:41

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