Superconvergence and a Posteriori Error Estimates of a Local Discontinuous Galerkin Method for the Fourth-order Initial-boundary Value Problems Arising in Beam Theory

Abstract. In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k+1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(h) superconvergent at the roots of (k + 1)-degree Radau polynomials. Computational results indicate that global superconvergence holds for LDG solutions. We discuss how to apply our superconvergence results to construct efficient and asymptotically exact a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Our results are valid for arbitrary regular meshes and for P k polynomials with k ≥ 1, and for various types of boundary conditions.