International Journal of Numerical Analysis and Modeling
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(hk+2) 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 Pk polynomials with k ≥ 1, and for various types of boundary conditions.
Baccouch, Mahboub, "Superconvergence and a posteriori error estimates of a local discontinuous Galerkin method for the fourth-order initial-boundary value problems arising in beam theory" (2014). Mathematics Faculty Publications. 7.