Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem

TL;DR

This paper proves supercloseness and optimal error bounds for the local discontinuous Galerkin method applied to a singularly perturbed convection-diffusion problem with boundary layers, on layer-adapted meshes.

Contribution

It establishes supercloseness and optimal $L^2$ error bounds for LDG solutions on three types of layer-adapted meshes, advancing numerical analysis of singularly perturbed problems.

Findings

Supercloseness of order $O(N^{-(k+1)})$ in energy norm for LDG solutions.

Optimal $L^2$ error bounds of order $O(N^{-(k+1)})$ up to logarithmic factors.

Numerical experiments confirm theoretical superconvergence results.

Abstract

A singularly perturbed convection-diffusion problem posed on the unit square in $\mathbb{R}^2$, whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with piecewise polynomials of degree at most $k>0$ on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type.On Shishkin-type meshes this method is known to be no greater than $O(N^{-(k+1/2)})$ accurate in the energy norm induced by the bilinear form of the weak formulation, where $N$ mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish $O(N^{-(k+1)})$ energy-norm superconvergence on all three types of mesh for the difference between the LDG solution and a local Gauss-Radau projection of the exact solution into the finite element space. This supercloseness property implies a new $N^{-(k+1)}$ bound for the $L^2$ error between the LDG solution on each type of mesh and the exact solution of the problem; this bound is optimal (up to logarithmic factors). Numerical experiments confirm our theoretical results.