Convergence analysis of a weak Galerkin finite element method on a Bakhvalov-type mesh for a singularly perturbed convection-diffusion equation in 2D

TL;DR

This work addresses a 2D singularly perturbed convection-diffusion problem with a small diffusion parameter $\varepsilon$ and develops a weak Galerkin finite element method on a Bakhvalov-type mesh to resolve boundary and corner layers. The WG framework uses discontinuous approximations with weak gradient $\nabla_w$ and discrete weak convection divergence $\nabla_w^{b}$, yielding a coercive bilinear form $A$ with stabilization terms and a projection operator that commutes with $\nabla_w$. The authors prove an energy-norm error estimate of order $O(N^{-k})$ that is independent of $\varepsilon$, via an error equation for $e_N=\mathcal{Q}_N u-u_N$ and standard stability arguments, and support the theory with numerical experiments. The results demonstrate robust performance for convection-dominated regimes on Bakhvalov-type meshes, offering an effective approach for uniform accuracy when resolving boundary and corner layers.

Abstract

In this paper, we propose a weak Galerkin finite element method (WG) for solving singularly perturbed convection-diffusion problems on a Bakhvalov-type mesh in 2D. Our method is flexible and allows the use of discontinuous approximation functions on the meshe. An error estimate is devised in a suitable norm and the optimal convergence order is obtained. Finally, numerical experiments are given to support the theory and to show the efficiency of the proposed method.

Figures (1)

• Figure 1: A Bakhvalov-type mesh with $N = 8$ and dissection of $\Omega$.

Theorems & Definitions (18)

• Lemma 2.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1
• proof