Local Discontinuous Galerkin Method for a Third-Order Singularly Perturbed Problem of Convection-Diffusion Type

Abstract

The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost O ⁢ ( N - ( k + 1 2 ) ) {O(N^{-(k+\frac{1}{2})})} (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, k ≥ 0 {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.