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 2D problems 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：针对定义在$\mathbb{R}^2$中单位正方形上的奇异摄动反应-扩散问题，采用局部间断伽辽金（LDG）有限元方法进行数值求解。此类二维问题的典型解在区域边界附近呈现边界层结构，这些边界层通常会给数值方法带来困难。本文提出的LDG方法通过采用Shishkin网格并引入"层迎风通量"的新概念来处理边界层——该离散通量在位于边界层内部的细网格上，其取值方向沿着边界层衰减的方向；在粗网格上则可使用标准的中心通量。与许多其他LDG变体不同，采用这些通量时无需添加惩罚项。我们所选的离散通量使得在平衡范数下进行最优阶误差分析成为可能；该范数比常规能量范数更强，是衡量奇异摄动反应-扩散问题数值解误差的更合适度量。理论证明表明，在平衡范数下LDG方法通常具有$O((N^{-1}\ln N)^{k+1})$阶收敛速度，其中$N$为每个坐标方向的网格区间数，且LDG方法在每个坐标变量上采用$k$次张量积分片多项式。该结果是LDG方法应用于此类问题的首例平衡范数误差估计，且在Shishkin网格上具有最优收敛性。数值实验验证了该估计的尖锐性。