Supercloseness analysis of the nonsymmetric interior penalty Galerkin method on Bakhvalov-type mesh

In this paper, we study the convergence of the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh for the first time. For this purpose, a new composite interpolation is designed, which solves the inherent difficulty of analysis on Bakhvalov-type meshes. More specifically, Gauss Radau interpolation and Gauss Lobatto interpolation are used outside and inside the layer, respectively. On the basis of that, by choosing the specific values of the penalty parameters at different mesh points, we derive the supercloseness of $k+1\mathrm{/}2$th order ($k\ge 1$), and prove the convergence of optimal order in an energy norm. The theoretical conclusion is consistent with the numerical results.