Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem

Abstract

A singularly perturbed convection-diffusion problem posed on the unit square in R 2 \mathbb {R}^2 , whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with tensor-product piecewise polynomials of degree at most k > 0 k>0 on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type. On Shishkin-type meshes this method is known to be no greater than O ( N − ( k + 1 / 2 ) ) O(N^{-(k+1/2)}) accurate in the energy norm induced by the bilinear form of the weak formulation, where N N mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish O ( N − ( k + 1 ) ) O(N^{-(k+1)}) energy-norm superconvergence on all three types of mesh for the difference between the LDG solution and a local Gauss-Radau projection of the true solution into the finite element space. This supercloseness property implies a new N − ( k + 1 ) N^{-(k+1)} bound for the L 2 L^2 error between the LDG solution on each type of mesh and the true solution of the problem; this bound is optimal (up to logarithmic factors). Numerical experiments confirm our theoretical results.