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Superconvergence of the numerical traces of discontinuous Galerkin and Hybridized methods for convection-diffusion problems in one space dimension

Authors: Fatih Celiker and Bernardo Cockburn
Journal: Math. Comp. 76 (2007), 67-96
MSC (2000): Primary 65M60, 65N30, 35L65
Published electronically: August 7, 2006
MathSciNet review: 2261012
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Abstract: In this paper, we uncover and study a new superconvergence property of a large class of finite element methods for one-dimensional convection-diffusion problems. This class includes discontinuous Galerkin methods defined in terms of numerical traces, discontinuous Petrov-Galerkin methods and hybridized mixed methods. We prove that the so-called numerical traces of both variables superconverge at all the nodes of the mesh, provided that the traces are conservative, that is, provided they are single-valued. In particular, for a local discontinuous Galerkin method, we show that the superconvergence is order $ 2\,p+1$ when polynomials of degree at most $ p$ are used. Extensive numerical results verifying our theoretical results are displayed.

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Additional Information

Fatih Celiker
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455

Received by editor(s): May 12, 2005
Published electronically: August 7, 2006
Additional Notes: The second author was partially supported by the National Science Foundation (Grant DMS-0411254) and by the Minnesota Supercomputing Institute.
Article copyright: © Copyright 2006 American Mathematical Society

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