Superconvergence of the local discontinuous Galerkin method for linear fourth-order time-dependent problems in one space dimension

In this paper we investigate the superconvergence of local discontinuous Galerkin (LDG) methods for solving one-dimensional linear time-dependent fourth-order problems. We prove that the error between the LDG solution and a particular projection of the exact solution, ēu , achieves ( k+ 2 ) th-order superconvergence when polynomials of degree k (k 1) are used. Numerical experiments with Pk polynomials, with 1 k 3, are displayed to demonstrate the theoretical results, which show that the error ēu actually achieves (k +2)th-order superconvergence, indicating that the error bound for ēu obtained in this paper is suboptimal. Initial boundary value problems, nonlinear equations and solutions having singularities, are numerically investigated to verify that the conclusions hold true for very general cases.