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An Optimal Estimate for the Local Discontinuous Galerkin Method

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Discontinuous Galerkin Methods

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 11))

Abstract

L 2 error estimates for the Local Discontinuous Galerkin (LDG) method have been theoretically proven for linear convection diffusion problems and periodic boundary conditions. It has been proven that when polynomials of degree k are used, the LDG method has a suboptimal order of convergence k. However, numerical experiments show that under a suitable choice of the numerical flux, higher order of convergence can be achieved. In this paper, we consider Dirichlet boundary conditions and we show that the LDG method has an optimal order of convergence k + 1.

Partially supported by National Science Foundation grant DMS-9805617 and by the University of Minnesota Supercomputer Institute

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References

  1. Cockburn, B., Shu, C.W.: The Local Discontinuous Galerkin Finite Element Method for convection diffusion systems. SIAM J. Numer. Anal. 35, (1998) 2440–2463

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  2. CSK] Cockburn, B., Shu, C.W., Karniadakis, G.E.: An overview of the development of DG methods. This volume, Springer Verlag, Berlin, Germany

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  3. LeSaint, P., Raviart, P.A.: On a finite element method for solving the neutron transport equation. In: Mathematical aspects of finite elements in partial differential equations. Academic Press, C.A de Boor (Ed), Academic Press, New York (1974), 89–145

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Authors and Affiliations

  1. Scientific Computation, University of Minnesota, Minneapolis, MN, 55455, USA

    Paul Castillo

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  1. Paul Castillo
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Editors and Affiliations

  1. School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN, 55455, USA

    Bernardo Cockburn

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    George E. Karniadakis  & Chi-Wang Shu  & 

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© 2000 Springer-Verlag Berlin Heidelberg

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Castillo, P. (2000). An Optimal Estimate for the Local Discontinuous Galerkin Method. In: Cockburn, B., Karniadakis, G.E., Shu, CW. (eds) Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59721-3_23

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  • DOI: https://doi.org/10.1007/978-3-642-59721-3_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64098-8

  • Online ISBN: 978-3-642-59721-3

  • eBook Packages: Springer Book Archive

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