Uniform convergence of optimal order under a balanced norm of a local discontinuous Galerkin method on a Shishkin mesh

TL;DR

This paper establishes optimal convergence of a local discontinuous Galerkin method under a balanced norm for singularly perturbed reaction-diffusion problems on Shishkin meshes, extending results from 1D to 2D.

Contribution

It introduces a new balanced norm, novel numerical fluxes, and a specialized interpolation to achieve optimal convergence in LDG methods for reaction-diffusion problems.

Findings

Optimal convergence under the balanced norm is achieved in 1D.

The method extends to 2D with similar convergence properties.

Numerical experiments confirm the theoretical convergence rates.

Abstract

For singularly perturbed reaction-diffusion problems in 1D and 2D, we study a local discontinuous Galerkin (LDG) method on a Shishkin mesh. In these cases, the standard energy norm is too weak to capture adequately the behavior of the boundary layers that appear in the solutions. To deal with this deficiency, we introduce a balanced norm stronger than the energy norm. In order to achieve optimal convergence under the balanced norm in one-dimensional case, we design novel numerical fluxes and propose a special interpolation that consists of a Gauss-Radau projection and a local $L^2$ projection. Moreover, we generalize the numerical fluxes and interpolation, and extend convergence analysis of optimal order from 1D to 2D. Finally, numerical experiments are presented to confirm the theoretical results.