The local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem with characteristic and exponential layers
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A singularly perturbed convection-diffusion problem,posed on the unit square in 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((N^-1ln N)^k+1/2) 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((N^-1ln N)^k+1) 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 (N^-1ln N)^k+1 for our LDG method. Numerical experiments show the sharpness of our theoretical results.
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