This paper investigates the supercloseness of a singularly perturbed convection diffusion problem using the direct discontinuous Galerkin (DDG) method on a Shishkin mesh. The main technical difficulties lie in controlling the diffusion term inside the layer, the convection term outside the layer, and the inter element jump term caused by the discontinuity of the numerical solution. The main idea is to design a new composite interpolation, in which a global projection is used outside the layer to satisfy the interface conditions determined by the selection of numerical flux, thereby eliminating or controlling the troublesome terms on the unit interface; and inside the layer, Gau{\ss} Lobatto projection is used to improve the convergence order of the diffusion term. On the basis of that, by selecting appropriate parameters in the numerical flux, we obtain the supercloseness result of almost $k+1$ order under an energy norm. Numerical experiments support our main theoretical conclusion.

翻译：本文研究采用直接间断伽辽金（DDG）方法在Shishkin网格上处理奇异摄动对流扩散问题的超逼近性。主要技术难点在于控制层内扩散项、层外对流项以及由数值解不连续性引起的单元间跳跃项。核心思想是设计一种新型复合插值：在层外使用全局投影以满足由数值通量选择确定的界面条件，从而消除或控制单元界面上的困难项；在层内采用Gau{\ss} Lobatto投影以提升扩散项的收敛阶数。在此基础上，通过选取数值通量中的适当参数，我们获得了在能量范数下几乎为$k+1$阶的超逼近结果。数值实验支持了我们的主要理论结论。