Local discontinuous Galerkin method for a third‐order singularly perturbed problem of reaction–diffusion type

Abstract

AbstractA third‐order singularly perturbed problem of reaction–diffusion type is solved numerically by the local discontinuous Galerkin (LDG) method. By using the local Gauss–Radau projection for the primal variable u, but local L2 projection for the two auxiliary variables p and q, we obtain an optimal convergence rate of order (or up to a logarithmic factor) in the energy norm, uniformly in the singular perturbation parameter. Here is the degree of the piecewise polynomials used in the discrete space and N is the number of mesh intervals. The layer‐adapted meshes are used including a standard Shishkin mesh, a Bakhvalov–Shishkin mesh, and a Bakhvalov‐type mesh. Numerical experiments are given to show the sharpness of our theoretical results.