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.

Hp-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes / Cangiani, A.; Dong, Z.; Georgoulis, E. H.. - In: SIAM JOURNAL ON SCIENTIFIC COMPUTING. - ISSN 1064-8275. - 39:4(2017), pp. 1251-1279. [10.1137/16M1073285]

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

Cangiani A.;Dong Z.;
2017-01-01

Abstract

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.
2017
39
4
1251
1279
https://arxiv.org/abs/1605.01212
Cangiani, A.; Dong, Z.; Georgoulis, E. H.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135245
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