Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

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 artificially 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 L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.

Supercloseness in a balanced norm of the NIPG method on Shishkin mesh for a reaction diffusion problem

“…In this paper, we mainly discuss singularly perturbed reaction-diffusion equations. For these reaction-diffusion problems, the authors in [12] have analyzed uniform convergence under an energy norm. But the energy norm is too weak to fully capture the behavior of the layers appearing in the solutions.…”