Inf-sup stable space-time Local Discontinuous Galerkin method for the heat equation

TL;DR

This work develops and analyzes a space-time Local Discontinuous Galerkin method for the heat equation on general prismatic space-time meshes. It proves well-posedness via inf-sup conditions without relying on polynomial inverse estimates, and it derives hp-a priori error estimates for several discrete spaces, including tensor-product, standard, quasi-Trefftz, and embedded Trefftz spaces. The paper demonstrates two inf-sup stability results, one general and one space of polynomial adherence with an additional time-derivative inclusion, and provides numerical evidence of optimal convergence rates and favorable conditioning, with substantial DoF reductions for Trefftz variants. Overall, the method offers a robust, high-order, flexible framework for parabolic problems, enabling efficient space-time discretization and potential gains in accuracy and performance through Trefftz-type spaces.

Abstract

We propose and analyze a space-time Local Discontinuous Galerkin method for the approximation of the solution to parabolic problems. The method allows for very general discrete spaces and prismatic space-time meshes. Existence and uniqueness of a discrete solution are shown by means of an inf-sup condition, whose proof does not rely on polynomial inverse estimates. Moreover, for piecewise polynomial spaces satisfying an additional mild condition, we show a second inf-sup condition that provides additional control over the time derivative of the discrete solution. We derive $hp$-a priori error bounds based on these inf-sup conditions, which we use to prove convergence rates for standard, tensor-product, and quasi-Trefftz polynomial spaces. Numerical experiments validate our theoretical results.

Figures (8)

• Figure 1: Example of a $(2+1)$-dimensional prismatic space--time mesh with polygonal space-like facets and hanging time-like facets.
• Figure 2: Example of a $(2+1)$-dimensional prismatic element. A space-like facet ($F^{{\mathrm{space}}}$) and a time-like facet ($F^{{\mathrm{time}}}$) are highlighted in gray and blue, respectively.
• Figure 3: Condition number of the stiffness matrix in $(1 + 1)$ dimensions for the four discrete spaces presented in Section \ref{['SECT::DISCRETE-SPACES']}. The numbers in the yellow boxes are the empirical algebraic rates.
• Figure 4: $h$-convergence for the $(2 + 1)$-dimensional problem with exact solution $u$ in \ref{['EQ::NUMEXP-SMOOTHSOL']}. The error is measured in the norms $|\!|\!|\cdot|\!|\!|_{_{\ref{['EQN::DG-NORMS-3']}}}$, $|\!|\!|\cdot|\!|\!|_{_{\ref{['EQN::DG-NORMS-4']}}}$, and $\|\cdot\|_{L^2(Q_T)}$, in the corresponding rows. The columns correspond to polynomial degree $p=2,3,4$. The numbers in the yellow boxes are the empirical algebraic convergence rates corresponding to the embedded Trefftz space.
• Figure 5: $p$-convergence in the norms $\|\cdot\|_{_{\ref{['EQN::DG-NORMS-3']}}}$ (left panel), $\|\cdot\|_{_{\ref{['EQN::DG-NORMS-4']}}}$ (central panel), and $\|\cdot\|_{L^2(Q_T)}$ (right panel) for the $(2+ 1)$-dimensional problem with exact solution $u$ in \ref{['EQ::NUMEXP-SMOOTHSOL']}.

Theorems & Definitions (51)

• Remark 2.2: Flux formulation
• Remark 2.3: Lifting operator
• Remark 2.4: Reduced formulation
• Lemma 3.2: LDG norm
• proof
• Lemma 3.3: Coercivity and continuity of $\ref{['DEF::Ah']}$
• proof
• Lemma 3.4: Coercivity of $\ref{['EQN::REDUCED-VARIATIONAL-DG']}$
• proof
• Theorem 3.5: Inf-sup stability