Uniform Convergence Analysis of the Discontinuous Galerkin Method on Layer-Adapted Meshes for Singularly Perturbed Problem

Abstract

This paper concerns a discontinuous Galerkin (DG) method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transformations. We derive an optimal convergence of the DG method for eight layer-adapted meshes in a general framework. The convergence rate is valid independent of the small parameter. Furthermore, we establish a sharperL2-error estimate if the true solution has a special regular component. Numerical experiments are also given.