SUPG-stabilized time-DG finite and virtual elements for the time-dependent advection-diffusion equation

TL;DR

The paper tackles robust, high-order time-DG discretizations for the time-dependent advection-diffusion equation using SUPG stabilization within a virtual element framework. It proves an inf-sup stability result with a constant independent of the mesh size $h$, time step $\tau$, and diffusion coefficient $\nu$, and derives a priori error bounds that are optimal in the energy norm across convection- and diffusion-dominated regimes without introducing a reactive term. The analysis encompasses both standard finite elements and virtual element spaces, including Serendipity VE variants, and yields stability and convergence results valid in $L^2(0,T;L^2(\Omega))$ without problem modification. Numerical experiments in $(3+1)$ dimensions validate the theory, demonstrating robust performance, especially in convection-dominated cases where the energy-norm error decays like $\mathcal{O}(h^{k+1/2})$ for polynomial degree $k$. Overall, the work provides a rigorous, high-order, artificially stabilized method that remains accurate and stable for challenging advection-diffusion problems on general polyhedral meshes, with practical implications for complex fluid-structure and multiphysics simulations.

Abstract

We carry out a stability and convergence analysis for the fully discrete scheme obtained by combining a finite or virtual element spatial discretization with the upwind-discontinuous Galerkin time-stepping applied to the time-dependent advection-diffusion equation. A space-time streamline-upwind Petrov-Galerkin term is used to stabilize the method. More precisely, we show that the method is inf-sup stable with constant independent of the diffusion coefficient, which ensures the robustness of the method in the convection- and diffusion-dominated regimes. Moreover, we prove optimal convergence rates in both regimes for the error in the energy norm. An important feature of the presented analysis is the control in the full $L^2(0,T;L^2(Ω))$ norm without the need of introducing an artificial reaction term in the model. We finally present some numerical experiments in $(3 + 1)$-dimensions that validate our theoretical results.

Figures (4)

• Figure 1: The interior of some spatial meshes of each type with approximately the same meshsize.
• Figure 2: Convergence Analysis: the trend of all errors taken into account in the diffusion-dominated regime, i.e., for $\nu=1$.
• Figure 3: Convergence Analysis: the trend of all errors taken into account in the convection-dominated regime, i.e., for $\nu=10^{-10}$.
• Figure 4: Benchmark problem: comparison between the discrete solutions obtained with the schemes NONE and SUPG at different time steps.

Theorems & Definitions (46)

• Remark 3.2: Relaxation of the mesh assumptions
• Remark 3.3: Stability terms
• Remark 3.4
• Theorem 3.6: Inf-sup stability
• Corollary 3.7: Well-posedness
• Remark 3.8: Stability in $L^2(0,T;L^2(\Omega))$
• Theorem 3.10: A priori error estimates
• Lemma 4.1: Local inverse estimates
• proof
• Lemma 4.2