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

TL;DR

This paper investigates the supercloseness properties of the NIPG method on Shishkin meshes for singularly perturbed convection-diffusion problems, introducing new analysis techniques and post-processing to enhance solution accuracy.

Contribution

The paper presents a novel supercloseness analysis for NIPG on Shishkin meshes, using specialized interpolations and penalty parameters, and introduces a post-processing method for higher accuracy.

Findings

Supercloseness order nearly k+1 under energy norm.

Post-processing improves numerical solution accuracy.

Numerical results confirm theoretical superconvergence.

Abstract

In this paper, we analyze the supercloseness result of nonsymmetric interior penalty Galerkin (NIPG) method on Shishkin mesh for a singularly perturbed convection diffusion problem. According to the characteristics of the solution and the scheme, a new analysis is proposed. More specifically, Gau{\ss} Lobatto interpolation and Gau{\ss} Radau interpolation are introduced inside and outside the layer, respectively. By selecting special penalty parameters at different mesh points, we further establish supercloseness of almost k + 1 order under the energy norm. Here k is the order of piecewise polynomials. Then, a simple post processing operator is constructed. In particular, a new analysis is proposed for the stability analysis of this operator. On the basis of that, we prove that the corresponding post-processing can make the numerical solution achieve higher accuracy. Finally, superconvergence can be derived under the discrete energy norm. These theoretical conclusions can be verified numerically.