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We present a new $hp$-version space-time discontinuous Galerkin (dG) finiteelement method for the numerical approximation of parabolic evolution equationson general spatial meshes consisting of polygonal/polyhedral (polytopic)elements, giving rise to prismatic space-time elements. A key feature of theproposed method is the use of space-time elemental polynomial bases of\emph{total} degree, say $p$, defined in the physical coordinate system, asopposed to standard dG-time-stepping methods whereby spatial elemental basesare tensorized with temporal basis functions. This approach leads to a fullydiscrete $hp$-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 keyfeature of the new space-time dG method is the incorporation of very generalspatial meshes consisting of possibly polygonal/polyhedral elements with\emph{arbitrary} number of faces. A priori error bounds are shown for theproposed method in various norms. An extensive comparison among the newspace-time dG method, the (standard) tensorized space-time dG methods, theclassical dG-time-stepping, and conforming finite element method in space, ispresented in a series of numerical experiments.

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