Optimal balanced-norm error estimate of the LDG method for reaction-diffusion problems II: the two-dimensional case with layer-upwind flux

TL;DR

This paper introduces a novel LDG finite element method with a layer-upwind flux for 2D reaction-diffusion problems, achieving optimal error estimates on Shishkin meshes despite boundary layers.

Contribution

It develops a new LDG method with layer-upwind flux and proves optimal error estimates in a balanced norm for 2D reaction-diffusion problems with boundary layers.

Findings

Achieves convergence order of O((N^{-1}ln N)^{k+1}) in the balanced norm.

Introduces a layer-upwind flux that handles boundary layers without penalty terms.

Numerical experiments confirm the theoretical optimality of the error estimates.

Abstract

A singularly perturbed reaction-diffusion problem posed on the unit square in $\mathbb{R}^2$ is solved numerically by a local discontinuous Galerkin (LDG) finite element method. Typical solutions of this class of 2D problems exhibit boundary layers along the sides of the domain; these layers generally cause difficulties for numerical methods. Our LDG method handles the boundary layers by using a Shishkin mesh and also introducing the new concept of a ``layer-upwind flux" -- a discrete flux whose values are chosen on the fine mesh (which lies inside the boundary layers) in the direction where the layer weakens. On the coarse mesh, one can use a standard central flux. No penalty terms are needed with these fluxes, unlike many other variants of the LDG method. Our choice of discrete flux makes it feasible to derive an optimal-order error analysis in a balanced norm; this norm is stronger than the usual energy norm and is a more appropriate measure for errors in computed solutions for singularly perturbed reaction-diffusion problems. It will be proved that the LDG method is usually convergent of order $O((N^{-1}\ln N)^{k+1})$ in the balanced norm, where $N$ is the number of mesh intervals in each coordinate direction and tensor-product piecewise polynomials of degree~$k$ in each coordinate variable are used in the LDG method. This result is the first of its kind for the LDG method applied to this class of problem and is optimal for convergence on a Shishkin mesh. Its sharpness is confirmed by numerical experiments.