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

TL;DR

This paper introduces a novel $hp$-version space-time discontinuous Galerkin method for parabolic problems on complex meshes, reducing degrees of freedom while maintaining accuracy, and compares it with existing methods through numerical experiments.

Contribution

It proposes a new space-time dG scheme using polynomial bases of total degree in physical coordinates, applicable to general polygonal/polyhedral meshes, with proven error bounds.

Findings

Reduces degrees of freedom compared to standard methods.

Achieves acceptable approximation despite less degrees of freedom.

Demonstrates effectiveness through numerical comparisons.

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 \emph{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 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.