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

Abstract

Some popular stabilization techniques, such as nonsymmetric interior penalty Galerkin (NIPG) method, have important application value in computational fluid dynamics. In this paper, we analyze a NIPG method on Shishkin mesh for a singularly perturbed convection diffusion problem, which is a typical simplified fluid model. According to the characteristics of the solution, the mesh and the numerical scheme, a new interpolation is designed for convergence analysis. More specifically, Gauß Lobatto interpolation and Gauß Radau interpolation are introduced inside and outside the layer, respectively. On the basis of that, by selecting special penalty parameters at different mesh points, we establish supercloseness of almost $k+1$ order in an energy norm. Here $k\ge 1$ is the degree of piecewise polynomials. Then, a simple post-processing operator is constructed, and it is proved that the corresponding post-processing can make the numerical solution achieve higher accuracy. In this process, a new analysis is proposed for the stability analysis of this operator. Finally, superconvergence is derived under a discrete energy norm. These conclusions can be verified numerically. Furthermore, numerical experiments show that the increase of polynomial degree k and mesh parameter N, the decrease of perturbation parameter $\varepsilon$ or the use of over-penalty technology may increase the condition number of linear system. Therefore, we need to cautiously consider the application of high-order algorithm.