Local discontinuous Galerkin method for a singularly perturbed fourth-order problem of reaction–diffusion type

Abstract

A fourth-order singularly perturbed problem of reaction–diffusion type is solved numerically by a local discontinuous Galerkin (LDG) method. Under suitable hypotheses, we prove optimal convergence of the LDG method on a Shishkin mesh; that is, when piecewise polynomials of degree k are used on a Shishkin mesh with N mesh intervals, we obtain O((N−1lnN)k+1/2) convergence in the energy norm. The error bound is uniformly valid with respect to the singular perturbation parameter. In the error analysis, we exploit a relationship between the numerical solution of the third-order derivative with the gradient, the numerical solution and its element interface jump. We discuss also the convergence of the LDG method on two Bakhvalov-type meshes. Numerical experiments indicate that our error estimate is sharp.