Volume 5, Issue 3
Superconvergence and a Posteriori Error Estimates of a Local Discontinuous Galerkin Method

Zichen Deng

Int. J. Numer. Anal. Mod. B, 5 (2014), pp. 188-216

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  • 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^{k+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 P^k polynomials with k ≥ 1, and for various types of boundary conditions.

  • History

Published online: 2014-05

  • AMS Subject Headings

65M60, 65N30, 74K10

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