Local Error Estimates of the Ldg Method for 1-d Singularly Perturbed Problems

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 error of the approximations to the solution and its derivative converges at the optimal rate O(h) 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.