The local discontinuous Galerkin method on layer-adapted meshes for time-dependent singularly perturbed convection-diffusion problems

TL;DR

This paper analyzes the error of a local discontinuous Galerkin method combined with layer-adapted meshes and implicit time-stepping for solving time-dependent singularly perturbed convection-diffusion problems, providing uniform error estimates.

Contribution

It introduces a comprehensive error analysis for LDG methods on layer-adapted meshes for time-dependent problems, including uniform error bounds and improved estimates in 1D.

Findings

Uniform error estimates of order k+1/2 in space

Optimal error estimates in time

Numerical experiments confirm theoretical results

Abstract

In this paper we analyze the error as well for the semi-discretization as the full discretization of a time-dependent convection-diffusion problem. We use for the discretization in space the local discontinuous Galerkin (LDG) method on a class of layer-adapted meshes including Shishkin-type and Bakhvalov-type meshes and the implicit $\theta$-scheme in time. For piecewise tensor-product polynomials of degree $k$ we obtain uniform or almost uniform error estimates with respect to space of order $k+1/2$ in some energy norm and optimal error estimates with respect to time. Our analysis is based on careful approximation error estimates for the Ritz projection related to the stationary problem on the anisotropic meshes used. We discuss also improved estimates in the one-dimensional case and the use of a discontinuous Galekin discretization in time. Numerical experiments are given to support our theoretical results.