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.

翻译：作为一类流行的稳定化技术，非对称内罚Galerkin（NIPG）方法在计算流体力学中具有重要的应用价值。本文针对二维Shishkin网格上典型奇异扰动对流扩散问题，研究了NIPG方法。基于解、网格及数值格式的特征，引入了一种新型复合插值。事实上，该插值由层内顶点-边-单元插值和层外局部$L^{2}$投影构成。在此基础上，通过选取不同类型单元边上的罚参数，进一步在能量范数下获得了接近$k+\frac{1}{2}$阶的超逼近性质，其中$k$为分片多项式次数。数值实验验证了我们的理论结论。