Optimal maximum-norm estimate of the LDG method for singularly perturbed convection–diffusion problem

Abstract

We consider the local discontinuous Galerkin (LDG) method with a generalized alternating numerical flux for a one-dimensional singularly perturbed convection–diffusion problem. The double-optimal local maximum-norm error estimate is derived on the quasi-uniform meshes for the first time. Here, “double-optimal” refers to the maximum-norm error estimate out of the boundary layer region being optimal and the width of the boundary layer being quasi-optimal. Furthermore, we improve the discrete maximum principle and global L1-error estimate established in the literature.