The local discontinuous Galerkin method for a singularly perturbed convection–diffusion problem with characteristic and exponential layers

Abstract

A singularly perturbed convection–diffusion problem, posed on the unit square in ${\mathbb {R}}^2$, is studied; its solution has both exponential and characteristic boundary layers. The problem is solved numerically using the local discontinuous Galerkin (LDG) method on Shishkin meshes. Using tensor-product piecewise polynomials of degree at most $k>0$ in each variable, the error between the LDG solution and the true solution is proved to converge, uniformly in the singular perturbation parameter, at a rate of $O\left( \left( N^{-1}\ln N\right) ^{k+1/2}\right)$ in an associated energy norm, where N is the number of mesh intervals in each coordinate direction. (This is the first uniform convergence result proved for the LDG method applied to a problem with characteristic boundary layers.) Furthermore, we prove that this order of convergence increases to $O\left( \left( N^{-1}\ln N\right) ^{k+1}\right)$ when one measures the energy-norm difference between the LDG solution and a local Gauss-Radau projection of the true solution into the finite element space. This uniform supercloseness property implies an optimal $L^2$ error estimate of order $\left( N^{-1}\ln N\right) ^{k+1}$ for our LDG method. Numerical experiments show the sharpness of our theoretical results.