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 paper introduces a weak Galerkin finite element method tailored for 2D singularly perturbed convection-diffusion equations on Bakhvalov-type meshes, providing optimal convergence and validated by numerical experiments.

Contribution

It develops a flexible weak Galerkin method with proven optimal convergence for singularly perturbed problems on specialized meshes.

Findings

Optimal convergence order achieved

Method effectively handles singular perturbations

Numerical results confirm theoretical predictions

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.