Local discontinuous Galerkin method on layer-adapted meshes for singularly perturbed reaction-diffusion problems in two dimensions

TL;DR

This paper analyzes the LDG method for 2D singularly perturbed reaction-diffusion problems on layer-adapted meshes, establishing uniform and optimal convergence rates with theoretical error estimates and numerical validation.

Contribution

It introduces a comprehensive analysis of LDG on layer-adapted meshes, deriving error estimates in energy and balanced norms, and demonstrates uniform convergence and optimal rates.

Findings

Uniform convergence of order k in the balanced norm.

Optimal convergence of order k+1 in the energy norm.

Numerical experiments confirm theoretical results.

Abstract

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction-diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and "balanced" norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the "balanced" norm is artifically introduced to capture the boundary layer contribution. We establish a uniform convergence of order $k$ for the LDG method using the balanced norm with the local weighted $L^2$ projection as well as an optimal convergence of order $k+1$ for the energy norm using the local Gauss-Radau projections. Numerical experiments are presented.