A singularly perturbed reaction-diffusion problem posed on the unit square in $\mathbb{R}^2$ is solved numerically by a local discontinuous Galerkin (LDG) finite element method. Typical solutions of this class of problem exhibit boundary layers along the sides of the domain; these layers generally cause difficulties for numerical methods. Our LDG method handles the boundary layers by using a Shishkin mesh and also introducing the new concept of a ``layer-upwind flux" -- a discrete flux whose values are chosen on the fine mesh (which lies inside the boundary layers) in the direction where the layer weakens. On the coarse mesh, one can use a standard central flux. No penalty terms are needed with these fluxes, unlike many other variants of the LDG method. Our choice of discrete flux makes it feasible to derive an optimal-order error analysis in a balanced norm; this norm is stronger than the usual energy norm and is a more appropriate measure for errors in computed solutions for singularly perturbed reaction-diffusion problems. It will be proved that the LDG method is usually convergent of order $O((N^{-1}\ln N)^{k+1})$ in the balanced norm, where $N$ is the number of mesh intervals in each coordinate direction and tensor-product piecewise polynomials of degree~$k$ in each coordinate variable are used in the LDG method. This result is the first of its kind for the LDG method applied to this class of problem and is optimal for convergence on a Shishkin mesh. Its sharpness is confirmed by numerical experiments.

翻译：在$\mathbb{R}^2$中的单位正方形上提出的奇异摄动反应扩散问题，通过局部间断伽辽金（LDG）有限元方法进行数值求解。这类问题的典型解沿区域边界呈现边界层，这些边界层通常给数值方法带来困难。我们的LDG方法采用Shishkin网格处理边界层，并引入了"层迎流通量"的新概念——该离散通量的值在精细网格（位于边界层内部）上沿层衰减方向选取，而在粗网格上则可使用标准中心通量。与LDG方法的其他变体不同，这些通量无需添加罚项。这种离散通量的选择使得我们能够在平衡范数下推导出最优阶误差分析；该范数比常规能量范数更强，是衡量奇异摄动反应扩散问题计算解误差的更合适指标。本文将证明，在平衡范数下LDG方法的收敛阶通常为$O((N^{-1}\ln N)^{k+1})$，其中$N$为每个坐标方向的网格区间数，且LDG方法使用每个坐标变量上次数为$k$的张量积分段多项式。该结果首次针对此类问题的LDG方法建立，且对于Shishkin网格上的收敛性而言是最优的。数值实验证实了其尖锐性。