Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem
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A singularly perturbed convection-diffusion problem posed on the unit square in ℝ^2, whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with piecewise polynomials of degree at most 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)) 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 exact solution into the finite element space. This supercloseness property implies a new N^-(k+1) bound for the L^2 error between the LDG solution on each type of mesh and the exact solution of the problem; this bound is optimal (up to logarithmic factors). Numerical experiments confirm our theoretical results.
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