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

Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis

TL;DR

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.

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.

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

TL;DR

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.

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.

Paper Structure

This paper contains 17 sections, 10 theorems, 82 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem and method
  3. Model problem
  4. Finite element spaces
  5. Trace operators
  6. Space-time dG method
  7. Stability
  8. Inverse estimates
  9. Coercivity and continuity
  10. Inf-sup condition
  11. A priori error analysis
  12. Polynomial approximation
  13. Error-analysis
  14. Numerical examples
  15. Example 1
  16. ...and 2 more sections

Key Result

Lemma 4.1

\newlabellemmal_inv Let $\kappa_n \in \mathcal{U} \times \mathcal{T}$, and let Assumption A1 to hold. Then, for each $v\in\mathcal{P}_{p_{\kappa_n}}(\kappa_n)$, we have with $\mathcal{F}^\parallel = I_n\times F$, $F\subset\partial \kappa\cap \partial s^F_\kappa$ and $s^F_\kappa$ as in Assumption A1 sharing $F$ with $\kappa$.

Figures (9)

  • Figure 2.1: 30-gon with $\cup_{i=1}^{30} \bar{s}_\kappa^i = \bar{\kappa}$ (left); star shaped polygon with $\cup_{i=1}^{10} \bar{s}_\kappa^i \subsetneq \bar{\kappa}$ (right).
  • Figure 2.2: (a). $16$ polygonal spatial elements over the spatial domain $\Omega =(0,1)^2$; (b) space-time elements over $I_n \times \Omega$ under the local time partition $\mathcal{U}_n(\mathcal{T})$.
  • Figure 5.1: (a). Polygonal spatial element $\kappa$ and covering ${\@fontswitch{}{\mathcal{}} K}$; (b) space-time element $\kappa_n=I_n\times\kappa$ and covering ${\@fontswitch{}{\mathcal{}} K}_n:=I_n\times{\@fontswitch{}{\mathcal{}} K}$.
  • Figure 6.1: Example 1. DG(P) under $h$--refinement (left) and comparison with other methods (right) for three different norms.
  • Figure 6.2: Example 1. Convergence under $p$--refinement for $T=1$ with $80$ time steps (left) and for $T=40$ with $3200$ time steps (right) for three different norms. For (left) figures, DG(P) with $p=1,\dots,9$, DG(PQ) with $p=1,\dots,8$, DG(Q) and FEM(Q) with $p=1,\dots,7$.
  • ...and 4 more figures

Theorems & Definitions (24)

  • Remark 2.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Remark 4.4
  • Theorem 4.5
  • proof
  • ...and 14 more