SUPERCLOSENESS OF THE LOCAL DISCONTINUOUS GALERKIN METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM

A singularly perturbed convection-diffusion problem posed on the unit square in Double-struck capital R2, 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 on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalovtype. On Shishkin-type meshes this method is known to be no greater than O(N-(k+1/2)) accurate in the energy norm induced by the bilinear form of the weak formulation, where 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)) 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) bound for the L2 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.

Automated summary

This paper numerically solves a convection-diffusion problem on the unit square in Double-struck capital R2 with exponential boundary layers using the local discontinuous Galerkin (LDG) method. They establish the superconvergence property of the LDG solution on three types of layer-adapted meshes, which leads to an optimal bound for the L2 error. Numerical experiments confirm their theoretical results.