$hp$-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes
Abstract
We present a new -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 \emph{total} degree, say , 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 -dG scheme using less degrees of freedom for each time step, compared to standard dG time-stepping schemes employing tensorized space-time, 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 \emph{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.
Cite
@article{arxiv.1605.01212,
title = {$hp$-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes},
author = {Andrea Cangiani and Zhaonan Dong and Emmanuil H. Georgoulis},
journal= {arXiv preprint arXiv:1605.01212},
year = {2024}
}