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Optimal a priori error estimates for the $hp$-version of the local discontinuous Galerkin method for convection-diffusion problems


Authors: Paul Castillo, Bernardo Cockburn, Dominik Schötzau and Christoph Schwab
Journal: Math. Comp. 71 (2002), 455-478
MSC (2000): Primary 65N30; Secondary 65M60
DOI: https://doi.org/10.1090/S0025-5718-01-01317-5
Published electronically: May 11, 2001
MathSciNet review: 1885610
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Abstract:

We study the convergence properties of the $hp$-version of the local discontinuous Galerkin finite element method for convection-diffusion problems; we consider a model problem in a one-dimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh-width $h$, in the polynomial degree $p$, and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both $h$ and $p$. The theoretical results are confirmed in a series of numerical examples.


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Additional Information

Paul Castillo
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
Email: castillo@math.umn.edu

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Dominik Schötzau
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
Email: schoetza@math.umn.edu

Christoph Schwab
Affiliation: Seminar of Applied Mathematics, ETHZ, 8092 Zürich, Switzerland
Email: schwab@sam.math.ethz.ch

DOI: https://doi.org/10.1090/S0025-5718-01-01317-5
Keywords: Discontinuous Galerkin methods, $hp$-methods, convection-diffusion
Received by editor(s): February 21, 2000
Received by editor(s) in revised form: May 30, 2000
Published electronically: May 11, 2001
Additional Notes: The second author was partially supported by the National Science Foundation (Grant DMS-9807491) and by the University of Minnesota Supercomputer Institute. The third author was supported by the Swiss National Science Foundation (Schweizerischer Nationalfonds)
Article copyright: © Copyright 2001 American Mathematical Society

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