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 work addresses the numerical solution of a 2D singularly perturbed reaction-diffusion problem on the unit square using a local discontinuous Galerkin method on a Shishkin mesh. The authors introduce a layer-upwind flux, paired with a central flux on the coarse region, which allows a penalty-free LDG discretization and enables a rigorous balanced-norm error analysis. They prove near-optimal convergence $O((N^{-1}\ln N)^{k+1})$ in the balanced norm (and sharp energy-norm behavior) and confirm the results with numerical experiments that demonstrate the method's effectiveness in resolving boundary layers. The results represent the first optimal balanced-norm LDG error bound for this class of 2D problems on a Shishkin mesh, with implications for extending the approach to other layer-adapted meshes and singular perturbation problems.

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.

Figures (4)

• Figure 1: Division of $\Omega$ (left) and Shishkin mesh with $N=8$ (right).
• Figure 2: Vertex-edge-element approximation and three projectors $\mathcal{I}$, $\mathcal{J}$ and $\mathcal{K}$.
• Figure 3: Numerical solution $\mathbbm{U}_{64}$ and pointwise error $|u-\mathbbm{U}_{64}|$ in Example \ref{['exa:1']}. Here $k=2$.
• Figure 4: Numerical solution $\mathbbm{U}_{64}$ and pointwise error $|\mathbbm{U}_{64}-\widetilde{\mathbbm{U}}_{128}|$ in Example \ref{['exa:2']}. Here $k=2$.

Theorems & Definitions (28)

• Lemma 2.1
• Remark 2.2
• Remark 3.1
• Lemma 3.2
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof