Pointwise convergence of the local discontinuous Galerkin method on a Shishkin mesh for 2D reaction-diffusion problems

A local discontinuous Galerkin (LDG) method, based on a recently-developed “layer-upwind” flux, is applied on a Shishkin mesh to solve a reaction-diffusion problem posed on the 2D unit square. Typical solutions of this problem have an exponential boundary layer along each edge of the square. Energy-norm supercloseness, discrete Green’s functions, and local analysis with a cut-off function are combined to establish pointwise convergence for the LDG solution in the entire domain; in this derivation the coarse mesh, the highly anisotropic boundary layer mesh, and the fine corner mesh are each handled with a different technique to gain the sharpest local results. The convergence rates obtained are nearly optimal (provided that one disregards logarithmic factors and error terms that are multiplied by the small singular perturbation parameter) and are the best pointwise error bounds currently known for the LDG method when solving 2D reaction-diffusion problems with boundary layers. Numerical results are also included to test the sharpness of our results.