Local discontinuous Galerkin method for a third order singularly perturbed problem of convection-diffusion type

TL;DR

This paper analyzes the local discontinuous Galerkin method for a third-order singularly perturbed convection-diffusion problem, proving uniform convergence on various layer-adapted meshes and validating results with numerical experiments.

Contribution

It establishes almost optimal energy-norm convergence rates for LDG method on multiple layer-adapted meshes for a complex third-order perturbed problem.

Findings

Proves uniform convergence rates up to logarithmic factors.

Validates theoretical results with numerical experiments.

Applicable to Shishkin, Bakhvalov-Shishkin, and Bakhvalov meshes.

Abstract

The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of the convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost $O(N^{-(k+1/2)})$ (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, $k\geq 0$ is the maximum degree of piecewise polynomials used in discrete space, and $N$ is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin type, and Bakhvalov-type. Numerical experiments are conducted to test the theoretical results.