$hp$-Version Space-Time Discontinuous Galerkin Methods for Parabolic Problems on Prismatic Meshes

Abstract

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 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.Key words. space-time discontinuous Galerkin; hp-finite element methods; reduced cardinality basis functions; discontinuous Galerkin time-stepping.AMS subject classifications. 65N30, 65M60, 65J101. Introduction. The discontinuous Galerkin (dG) method can be traced back to [41], where it was introduced as a nonstandard finite element scheme for solving the neutron transport equation. This dG method was analyzed in [36], where it was also applied as a time stepping scheme for initial value problem for ordinary differential equations, and was shown to be equivalent to certain implicit Runge-Kutta methods. Jamet [34] introduced a dG time-stepping scheme for parabolic problems on evolving domains, later extended and analysed in [24,20,21,22,23]. For an introduction, we refer to the classic monograph [52] and the references therein. In [38], the quasioptimality of the dG time-stepping method for parabolic problems in mesh-dependent norms is established. Also, dG time-stepping convergence analyses under minimal regularity were shown in [57,14,15]. In all aforementioned literature, convergence of the discrete solution to the exact solution is achieved by reducing spatial mesh size h and time step size τ at some fixed (typically low) order.On the other hand, the p-and hp-version finite element method (FEM) appeared in the 1980s (see [7,6], and also the textbook [46] for a extensive survey). p-and hp-version FEM can achieve exponential rates of convergence when the underlying solution is locally analytic by increasing the polynomial order p and/or locally grading the meshsize towards corner or edge singularities. In this vein, the analyticity in the time-variable in parabolic problems has given rise to the use of p-and hp-versi...

Recovered finite element methods

“…where R r (T ) ∈ {P r (T ), Q r (T )}. We stress that, in this context, we can consider local bases in P r (T ) also for box-type elements; we shall return to this point below, cf., [11,10,9]. Further, let T + , T − be two (generic) elements sharing a facet e := ∂T + ∩ ∂T − ⊂ Γ int with respective outward normal unit vectors n + and n − on e. For a function v : Ω → R that may be discontinuous across Γ int , we set v + := v| e⊂∂T + , v − := v| e⊂∂T − , and we define the jump by…”

V-cycle Multigrid Algorithms for Discontinuous Galerkin Methods on Non-nested Polytopic Meshes

“…[1,2,3,4,6,8,14,15,16,27,33,38,46]. In particular, the efficient approach presented in [27] is based on defining a local polynomial discrete space by making use of the bounding box of each element [32]: this technique together with a careful choice of the discontinuity penalization parameter permits the use of polytopal elements which can be characterized by faces of arbitrarily small measure and as shown in [25], see also [6], possibly by an unbounded number of faces.…”

A high-order discontinuous Galerkin approach to the elasto-acoustic problem

“…Such a discretization is then performed using general polygonal or polyhedral (briefly, polytopic) elements, with no restriction on the number of faces each element can possess, and possibly allowing for face degeneration in mesh refinement. The dG method has been recently proven to successfully support polytopic meshes: we refer the reader, e.g., to [7,8,9,10,11,12,13,14,15], as well as to the comprehensive research monograph by Cangiani et al [16]. In addition to the dG method, several other methods are capable to support polytopic meshes, such as the Polygonal Finite Element method [17,18,19,20], the Mimetic Finite Difference method [21,22,23,24], the Virtual Element method [25,26,27,28], the Hybridizable Discontinuous Galerkin method [29,30,31,32,33], and the Hybrid High-Order method [34,35,36,37,38].…”