Supercloseness of weak Galerkin method for a singularly perturbed convection–diffusion problem in 2D

Abstract

In this paper we analyze a weak Galerkin method on a Shishkin rectangular mesh for a singularly perturbed convection–diffusion problem in two dimensions, whose solution has one exponential layer and two parabolic layers. The supercloseness of this method is achieved by a specially constructed interpolant. More specifically, the interpolant defined in the interior of each element consists of a vertices-edges-element interpolant inside the layers and a modified Gauß-Radau interpolant outside the layers, while the interpolant defined on the boundary of each element consists of a vertices-edges-element interpolant inside the layers and a weighted L2 projection outside the layers. Moreover, we make use of the over-penalization technique inside the layers, and then prove supercloseness of order k+1/2, even up to almost k+1 under appropriate assumptions. Numerical experiments support the theoretical result.