This talk is concerned with some recent progress on the local discontinuous Galerkin (LDG) method for the singularly perturbed problems. The LDG methods use the completely discontinuous polynomials as its discrete space. They are advantageous at the strong stability, high order accuracy, flexibility of hp adaptivity and local solvability. For different kinds of singuarly perturbed problems with boundary layers, we demonstrate that the LDG method on layer-adapted meshes can acheive optimal convergence and supercloseness uniformly in the singular perturbation parameter. We discuss both the well-known Shishkin-type mesh and Bakhvalov-type mesh. Numerical experiments are given to verify our theoretical results.

Dr. Yao Cheng received his PhD degree in Mathematics, Nanjing University, in 2016. He was an associate professor at School of Mathematical Sciences, Suzhou University of Science and Technology. His research mainly focuses on the local DG method for the singularly perturbed problems with layers, including the local analysis and uniform convergence analysis of the LDG method. In recent years, he has published more than ten articles on the international journals such as Math Comp, J Sci Comput, Numer Algor, Calcolo, J Comput Appl Math, Comput Math Appl., etc.