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

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journal contribution
posted on 15.03.2017, 16:01 by Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis
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 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 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 conforming finite element method in space, is presented in a series of numerical experiments.

Funding

AC acknowledges partial support from the EPSRC (Grant EP/L022745/1). EHG acknowledges support from The Leverhulme Trust (Grant RPG-2015-306).

History

Citation

SIAM Journal on Scientific Computing, 2017, 39(4), A1251–A1279

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

VoR (Version of Record)

Published in

SIAM Journal on Scientific Computing

Publisher

Society for Industrial and Applied Mathematics

Copyright date

2017

Available date

15/03/2017

Publisher version

http://epubs.siam.org/doi/abs/10.1137/16M1073285

Notes

AMS subject classifications. 65N30, 65M60, 65J10

Language

en