Convergence analysis of novel discontinuous Galerkin methods for a convection dominated problem

TL;DR

This work develops a numerically stable DG scheme for the convection-diffusion-reaction equation in the convection-dominated regime by employing the DG finite element differential calculus framework, with diffusion discretized via the dual-wind DG method and convection via an averaged discrete gradient complemented by stabilization. The authors prove optimal convergence orders in the convection-dominated limit, via both coercivity-based and inf-sup analyses for the reduced problem and the full problem, and provide projection-based error estimates that quantify the dependence on the diffusion parameter $\varepsilon$ and mesh size $h$. Numerical experiments in MATLAB corroborate the theory, showing absence of spurious oscillations, optimal interior convergence away from boundary/interior layers, and the influence of stabilization parameters. The results also reveal that the reduced problem is equivalent to a centered-flux DG method, and the framework offers a pathway to extensions such as optimal control problems constrained by convection-dominated PDEs.

Abstract

In this paper, we propose and analyze a numerically stable and convergent scheme for a convection-diffusion-reaction equation in the convection-dominated regime. Discontinuous Galerkin (DG) methods are considered since standard finite element methods for the convection-dominated equation cause spurious oscillations. We choose to follow a novel DG finite element differential calculus framework introduced in Feng et al. (2016) and approximate the infinite-dimensional operators in the equation with the finite-dimensional DG differential operators. Specifically, we construct the numerical method by using the dual-wind discontinuous Galerkin (DWDG) formulation for the diffusive term and the average discrete gradient operator for the convective term along with standard DG stabilization. We prove that the method converges optimally in the convection-dominated regime. Numerical results are provided to support the theoretical findings.

Figures (8)

• Figure 1: The operators $\mathcal{Q}_i^{\pm}$
• Figure 2: Results of Example \ref{['Ex:smsol']}: $u_h$ (left) and $u$ (right) for $h = 1/128$.
• Figure 3: Results of Example \ref{['ex:bl1']}: $u_h$ (left) and $u$ (right) for $\varepsilon = 10^{-9}$, $h = 1/128$
• Figure 4: Results of Example \ref{['ex:inl']}: $u_h$ (left) and the profile of $u_h$ at $x_1=0$ (right) for $\varepsilon=10^{-9}$ and $h = 1/128$.
• Figure 5: Results of Example \ref{['ex:inl']}: $u_h$ (left) and the profile of $u_h$ at $x_1=0$ (right) for $\varepsilon = 10^{-3}$ and $h = 1/128$.

Theorems & Definitions (29)

• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Lemma 3.1
• proof
• Remark 3.2
• Remark 3.3
• Lemma 3.4
• proof