Local Error Estimates of the LDG Method for 1-D Singularly Perturbed Problems

Abstract

In this paper local discontinuous Galerkin method (LDG) was analyzed for solving 1-D convection-diffusion equations with a boundary layer near the outflow boundary. Local error estimates are established on quasi-uniform meshes with maximum mesh size $h$. On a subdomain with $O(h\ln(1/h))$ distance away from the outflow boundary, the $L^2$ error of the approximations to the solution and its derivative converges at the optimal rate $O(h^{k+1})$ when polynomials of degree at most $k$ are used. Numerical experiments illustrate that the rate of convergence is uniformly valid and sharp. The numerical comparison of the LDG method and the streamline-diffusion finite element method are also presented.