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

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 $\epsilon$. In this paper, we investigate the theoretical aspect of this phenomenon under a simplified ODE model. In particular, we establish uniform convergence rates $\sqrt{\epsilon} \left( \frac{\ln N}{N} \right)^{k+1}$ for the $L^2$-norm and $\left (\frac{\ln N}{N} \right)^{2k+1}$ for the one-sided flux inside the boundary layer region. Here $N$ (even) is the number of elements.