Local discontinuous Galerkin method on layer-adapted meshes for singularly perturbed reaction-diffusion problems in two dimensions

Abstract: We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction-diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and "balanced" norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the "balanced" norm is artifically introduced to capture the boundary layer contribution. We establish a uniform convergence of order $k$ for the LDG method using the balanced norm with the local weighted $L^2$ projection as well as an optimal convergence of order $k+1$ for the energy norm using the local Gauss-Radau projections. Numerical experiments are presented.