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The local discontinuous Galerkin (LDG) method is studied for a third-ordersingularly perturbed problem of the convection-diffusion type. Based on aregularity assumption for the exact solution, we prove almost $O(N^{-(k+1/2)})$(up to a logarithmic factor) energy-norm convergence uniformly in theperturbation parameter. Here, $k\geq 0$ is the maximum degree of piecewisepolynomials used in discrete space, and $N$ is the number of mesh elements. Theresults are valid for the three types of layer-adapted meshes: Shishkin-type,Bakhvalov-Shishkin type, and Bakhvalov-type. Numerical experiments areconducted to test the theoretical results.

Local discontinuous Galerkin method for a third order singularly perturbed problem of convection-diffusion type