This article investigates a local discontinuous Galerkin (LDG) method for one-dimensional and two-dimensional singularly perturbed reaction-diffusion problems on a Shishkin mesh. During this process, due to the inability of the energy norm to fully capture the behavior of the boundary layers appearing in the solutions, a balanced norm is introduced. By designing novel numerical fluxes and constructing special interpolations, optimal convergences under the balanced norm are achieved in both 1D and 2D cases. Numerical experiments support the main theoretical conclusions.

翻译：本文研究了一维和二维奇异摄动反应扩散问题在Shishkin网格上的局部间断Galerkin（LDG）方法。由于能量范数无法完全捕捉解中出现的边界层行为，我们引入了一种平衡范数。通过设计新颖的数值通量并构造特殊的插值方法，在一维和二维情形下均实现了平衡范数下的最优收敛性。数值实验验证了主要理论结论。