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We analyse the local discontinuous Galerkin (LDG) method for two-dimensionalsingularly perturbed reaction-diffusion problems. A class of layer-adaptedmeshes, including Shishkin- and Bakhvalov-type meshes, is discussed within ageneral framework. Local projections and their approximation properties onanisotropic meshes are used to derive error estimates for energy and "balanced"norms. Here, the energy norm is naturally derived from the bilinear form of LDGformulation and the "balanced" norm is artifically introduced to capture theboundary 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 normusing the local Gauss-Radau projections. Numerical experiments are presented.

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