Supercloseness of the NIPG method for a singularly perturbed convection diffusion problem on Shishkin mesh in 2D

Abstract

As a popular stabilization technique, the nonsymmetric interior penalty Galerkin (NIPG) method has significant application value in computational fluid dynamics. In this paper, we study the NIPG method for a typical two-dimensional singularly perturbed convection diffusion problem on a Shishkin mesh. According to the characteristics of the solution, the mesh and numerical scheme, a new composite interpolation is introduced. In fact, this interpolation is composed of a vertices-edges-element interpolation within the layer and a local $L^{2}$-projection outside the layer. On the basis of that, by selecting penalty parameters on different types of interelement edges, we further obtain the supercloseness of almost $k+\frac{1}{2}$ order in an energy norm. Here $k$ is the degree of piecewise polynomials. Numerical tests support our theoretical conclusion.