The local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem with characteristic and exponential layers

TL;DR

This paper develops and analyzes a local discontinuous Galerkin method on Shishkin meshes for singularly perturbed convection-diffusion problems with boundary layers, proving uniform convergence and supercloseness results.

Contribution

It provides the first uniform convergence proof for LDG methods applied to problems with characteristic boundary layers, including supercloseness and optimal error estimates.

Findings

Uniform convergence rate of O((N^{-1} \, \ln N)^{k+1/2}) in energy norm.

Enhanced convergence rate of O((N^{-1} \, \ln N)^{k+1}) with a local Gauss-Radau projection.

Numerical experiments confirm theoretical sharpness.

Abstract

A singularly perturbed convection-diffusion problem,posed on the unit square in $\mathbb{R}^2$, is studied; its solution has both exponential and characteristic boundary layers. The problem is solved numerically using the local discontinuous Galerkin (LDG) method on Shishkin meshes. Using tensor-product piecewise polynomials of degree at most $k>0$ in each variable, the error between the LDG solution and the true solution is proved to converge, uniformly in the singular perturbation parameter, at a rate of $O((N^{-1}\ln N)^{k+1/2})$ in an associated energy norm, where $N$ is the number of mesh intervals in each coordinate direction.(This is the first uniform convergence result proved for the LDG method applied to a problem with characteristic boundary layers.) Furthermore, we prove that this order of convergence increases to $O((N^{-1}\ln N)^{k+1})$ when one measures the energy-norm difference between the LDG solution and a local Gauss-Radau projection of the true solution into the finite element space.This uniform supercloseness property implies an optimal $L^2$ error estimate of order $(N^{-1}\ln N)^{k+1}$ for our LDG method. Numerical experiments show the sharpness of our theoretical results.