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.

翻译：本文提出了一种新的$hp$版本时空间断Galerkin有限元方法，用于在由多边形/多面体单元构成的一般空间网格上数值逼近抛物型发展方程，从而形成棱柱形时空单元。该方法的一个关键特征是使用在物理坐标系中定义的、总次数为$p$的时空单元多项式基函数，这与标准dG时间步进方法形成对比——后者将空间单元基函数与时间基函数进行张量积。相较于采用张量化时空基函数的标准dG时间步进格式，此方法在每时间步使用更少的自由度，同时保持了可接受的逼近性能。新时空dG方法的第二个关键特征是能够容纳包含任意面数的多边形/多面体单元构成的极一般空间网格。我们在多种范数下证明了该方法具有先验误差界。通过一系列数值实验，系统比较了新时空dG方法、（标准）张量化时空dG方法、经典dG时间步进方法以及空间协调有限元方法的性能。