Mathematics of Computation

Optimal a priori error estimates for the

-version of the local discontinuous Galerkin method for convection-diffusion problems

Author(s): Paul Castillo; Bernardo Cockburn; Dominik Schötzau; Christoph Schwab.
Journal: Math. Comp. 71 (2002), 455-478.
MSC (2000): Primary 65N30; Secondary 65M60
Posted: May 11, 2001
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We study the convergence properties of the

-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

, in the polynomial degree

, and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both

and

. The theoretical results are confirmed in a series of numerical examples.

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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: 10.1090/S0025-5718-01-01317-5
PII: S 0025-5718(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
Posted: 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)
Copyright of article: Copyright 2001, American Mathematical Society

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