Uniform convergence of a weak Galerkin finite element method on Shishkin mesh for singularly perturbed convection-diffusion problems in 2D

Abstract

This paper presents a weak Galerkin finite element method, exploring polynomial approximations of various degree, for solving singularly perturbed convection-diffusion equation in 2D. On each mesh element, this method makes use of polynomials of degree k≥1 in the interior, polynomials of degree j≥1 on the boundary, vector-valued polynomials of degree l≥k−1 for the discrete weak gradient and polynomials of degree m≥k for the discrete weak convection divergence. Shishkin mesh is used in order that the method is uniformly convergent independent of the singular perturbation parameter. A special interpolation is delicately designed according to the structures of the designed method and Shishkin mesh. Then uniform convergence of optimal order is proved, as is also confirmed by the numerical experiments.