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Journal of Scientific Computing

, Volume 59, Issue 3, pp 795–840 | Cite as

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part I: Superconvergence Error Analysis

  • Mahboub Baccouch
Article

Abstract

In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler–Bernoulli beam equation in one space dimension. We prove the \(L^2\) stability of the scheme and several optimal \(L^2\) error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are \(\mathcal O (h^{k+3/2})\) super close to particular projections of the exact solutions for \(k\)th-degree polynomial spaces while computational results show higher \(\mathcal O (h^{k+2})\) convergence rate. Our proofs are valid for arbitrary regular meshes and for \(P^k\) polynomials with \(k\ge 1\), and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the \(L^2\)-norm under mesh refinement.

Keywords

Local discontinuous Galerkin method Fourth-order Euler–Bernoulli equation Superconvergence Projection Optimal error estimates Stability 

Notes

Acknowledgments

The author would also like to thank the anonymous referees for their constructive comments and remarks which helped improve the quality and readability of the paper. This research was partially supported by the NASA Nebraska Space Grant Program and UCRCA at the University of Nebraska at Omaha.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Nebraska at OmahaOmahaUSA

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