The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of the convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost $O(N^{-(k+1/2)})$ (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, $k\geq 0$ is the maximum degree of piecewise polynomials used in discrete space, and $N$ is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin type, and Bakhvalov-type. Numerical experiments are conducted to test the theoretical results.

翻译：本地不连续的 Galerkin (LDG) 方法是针对对流扩散类型的第三阶奇扰问题进行研究的。 根据对确切解决方案的常态假设,我们证明在扰动参数中能量- 中能量- 中枢趋同几乎( 直至对数系数) $O( N ⁇ - (k+1/2)}) $( 在对数系数) 一致。 这里, $k\ geq 0 美元是离散空间中使用的小片多球体的最大程度, $N是网状元素的数量。 结果适用于三种类型的层适应网状网状网状网形: Shishkin 类型、 Bakhvalov- Shishkin 类型和 Bakhvalov 类型。 进行了数字实验以测试理论结果 。