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A singularly perturbed convection-diffusion problem,posed on the unit squarein $\mathbb{R}^2$, is studied; its solution has both exponential andcharacteristic boundary layers. The problem is solved numerically using thelocal discontinuous Galerkin (LDG) method on Shishkin meshes. Usingtensor-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 $O((N^{-1}\lnN)^{k+1/2})$ in an associated energy norm, where $N$ is the number of meshintervals in each coordinate direction.(This is the first uniform convergenceresult proved for the LDG method applied to a problem with characteristicboundary layers.) Furthermore, we prove that this order of convergenceincreases to $O((N^{-1}\ln N)^{k+1})$ when one measures the energy-normdifference between the LDG solution and a local Gauss-Radau projection of thetrue solution into the finite element space.This uniform superclosenessproperty implies an optimal $L^2$ error estimate of order $(N^{-1}\ln N)^{k+1}$for our LDG method. Numerical experiments show the sharpness of our theoreticalresults.

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