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

Celiker, Fatih; Cockburn, Bernardo

doi:10.1090/S0025-5718-06-01895-3

Superconvergence of the numerical traces of discontinuous Galerkin and Hybridized methods for convection-diffusion problems in one space dimension
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by Fatih Celiker and Bernardo Cockburn;

Math. Comp. 76 (2007), 67-96

DOI: https://doi.org/10.1090/S0025-5718-06-01895-3

Published electronically: August 7, 2006

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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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