Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems

Abstract

Abstract. In this paper the local discontinuous Galerkin method (LDG) is considered for solving one-dimensional singularly perturbed two-point boundary value problems of convection-diffusion type and reaction-diffusion type. Error estimates are studied on Shishkin meshes. The L 2 error bounds for the LDG approximation of the solution and its derivative are uniformly valid with respect to the singular perturbation parameter. Numerical experiments indicate that the orders of convergence are sharp.Key words. Local discontinuous Galerkin method, singularly perturbed, Shishkin mesh.AMS subject classifications. 65L11, 65N30, 65N15. IntroductionThe discontinuous Galerkin (DG) method was introduced in 1973 as a way of solving the steady-state neutron transport equation [16]. Successively in 1974, Lesaint and Raviart made the first analysis for the linear advection equation [12]. Since then many DG methods were vigorously studied. Among these is the local discontinuous Galerkin (LDG) method, which was proposed in [3] and [9] by separating higher order operators into systems of first order equations so that classical DG methods can be extended to problems with second order operators, especially for convection-diffusion and hyperbolic equations. The state of the art of the development of these methods and their applications can be found in [1,2,5,7,10].In this work, we apply the LDG method to 1-D singularly perturbed problems with Dirichlet boundary conditions: