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An a posteriori error estimate for the variable-degree Raviart-Thomas method


Authors: Bernardo Cockburn and Wujun Zhang
Journal: Math. Comp. 83 (2014), 1063-1082
MSC (2010): Primary 65N15, 65N30
DOI: https://doi.org/10.1090/S0025-5718-2013-02789-5
Published electronically: October 31, 2013
MathSciNet review: 3167450
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Abstract: We propose a new a posteriori error analysis of the variable-degree, hybridized version of the Raviart-Thomas method for second-order elliptic problems on conforming meshes made of simplexes. We establish both the reliability and efficiency of the estimator for the $ L_2$-norm of the error of the flux. We also find the explicit dependence of the estimator on the order of the local spaces $ k\ge 0$; the only constants that are not explicitly computed are those depending on the shape-regularity of the simplexes. In particular, the constant of the local efficiency inequality is proven to behave like $ (k+{2})^{3/2}$. However, we present numerical experiments suggesting that such a constant is actually independent of $ k$.


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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Wujun Zhang
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: wujun@umd.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02789-5
Received by editor(s): April 6, 2011
Received by editor(s) in revised form: October 3, 2012
Published electronically: October 31, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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