A singularly perturbed convection–diffusion problem, posed on the unit square in \({\mathbb {R}}^2\), is studied; its solution has both exponential and characteristic boundary layers. The problem is solved numerically using the local discontinuous Galerkin (LDG) method on Shishkin meshes. Using tensor-product piecewise polynomials of degree at most \(k>0\) in each variable, the error between the LDG solution and the true solution is proved to converge, uniformly in the singular perturbation parameter, at a rate of \(O\left( \left( N^{-1}\ln N\right) ^{k+1/2}\right) \) in an associated energy norm, where N is the number of mesh intervals in each coordinate direction. (This is the first uniform convergence result proved for the LDG method applied to a problem with characteristic boundary layers.) Furthermore, we prove that this order of convergence increases to \(O\left( \left( N^{-1}\ln N\right) ^{k+1}\right) \) when one measures the energy-norm difference between the LDG solution and a local Gauss-Radau projection of the true solution into the finite element space. This uniform supercloseness property implies an optimal \(L^2\) error estimate of order \(\left( N^{-1}\ln N\right) ^{k+1}\) for our LDG method. Numerical experiments show the sharpness of our theoretical results.

研究了\({\mathbb {R}}^2\)中单位正方形上的奇摄动对流扩散问题；它的解同时具有指数边界层和特征边界层。使用 Shishkin 网格上的局部不连续 Galerkin (LDG) 方法对问题进行数值求解。在每个变量中使用次数最多为\(k>0\)的张量积分段多项式，证明 LDG 解与真实解之间的误差在奇异扰动参数中以 \( O \left( \left( N^{-1}\ln N\right) ^{k+1/2}\right) \)在相关的能量范数中，其中N是每个坐标方向的网格间隔数。（这是 LDG 方法应用于具有特征边界层问题的第一个一致收敛结果。）此外，我们证明了这个收敛阶增加到\(O\left( \left( N^{-1}\ ln N\right) ^{k+1}\right) \)当测量 LDG 解与真解在有限元空间中的局部 Gauss-Radau 投影之间的能量范数差时。对于我们的 LDG 方法，这种统一的超接近性属性意味着\(\left( N^{-1}\ln N\right) ^{k+1}\)阶的最优\(L^2\)误差估计。数值实验显示了我们理论结果的清晰度。