Optimal-order balanced-norm error estimate of the local discontinuous Galerkin method with alternating numerical flux for singularly perturbed reaction–diffusion problems

Abstract

Balanced-norm error bounds have been established in Cheng et al. (2022) for the local discontinuous Galerkin (LDG) method using alternating numerical flux on Shishkin-type meshes. However, the convergence rate is shown to be one-half order lower than the numerical results in the general case. This paper seeks to fill up this gap by introducing a new composite projector in the error analysis. We achieve an optimal-order error estimate in the balanced-norm for the LDG method on both Shishkin-type and Bakhvalov-type meshes, uniformly in the small perturbation parameter.