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$hp$-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes
journal contribution
posted on 2017-03-15, 16:01 authored by Andrea Cangiani, Zhaonan Dong, Emmanuil H. GeorgoulisWe 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–A1279Author affiliation
/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of MathematicsVersion
- VoR (Version of Record)