Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D

Abstract

It has been observed from the authors' numerical experiments (2007) that the Local Discontinuous Galerkin (LDG) method converges uniformly under the Shishkin mesh for singularly perturbed two-point boundary problems of the convection-diffusion type. Especially when using a piecewise polynomial space of degree k, the LDG solution achieves the optimal convergence rate k + 1 under the L 2-norm, and a superconvergence rate 2k +1 for the one-sided flux uniformly with respect to the singular perturbation parameter. In this paper, we investigate the theoretical aspect of this phenomenon under a simplified ODE model. In particular, we establish uniform convergence rates √ ln N N k+1 for the L 2-norm and ln N N 2k+1 for the one-sided flux inside the boundary layer region. Here N (even) is the number of elements.