Analysis of optimal superconvergence of a local discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

摘要

In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary value problems (BVPs) of the form u '' = f (x, u, u'), x is an element of [a, b] subject to some suitable boundary conditions at the endpoints x = a and x = b. We prove optimal L-2 error estimates for the solution and for the auxiliary variable that approximates the first order derivative. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gauss-Radau projections of the exact solutions. Moreover, we prove that the LDG solutions are superconvergent with order p + 2 toward Gauss-Radau projections of the exact solutions. Finally, we prove, for any polynomial degree p, the (2p + 1)th superconvergence rate of the LDG approximations at the upwind or downwind points and for the domain average under quasi-uniform meshes. Our numerical experiments demonstrate optimal rates of convergence and superconvergence. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p >= 1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results. (C) 2019 The Author(s). Published by Elsevier B.V. on behalf of IMACS.