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Contraction property of adaptive hybridizable discontinuous Galerkin methods


Authors: Bernardo Cockburn, Ricardo H. Nochetto and Wujun Zhang
Journal: Math. Comp. 85 (2016), 1113-1141
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/mcom/3014
Published electronically: August 17, 2015
MathSciNet review: 3454360
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Abstract: We establish the contraction property between consecutive loops of adaptive hybridizable discontinuous Galerkin methods for the Poisson problem with homogeneous Dirichlet condition. The contractive quantity is the sum of the square of the $ L^2$-norm of the flux error, which is not even monotone, and a two-parameter scaled error estimator, which quantifies both the lack of $ H(\mathrm {div},\Omega )$-conformity and the deviation from a gradient of the approximate flux. A distinctive and novel feature of this analysis, which enables comparison between two nested meshes, is the lifting of trace residuals from inter-element boundaries to element interiors.


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Additional Information

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

Ricardo H. Nochetto
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
Email: rhn@math.umd.edu

Wujun Zhang
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, and Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: wujun@math.umd.edu

DOI: https://doi.org/10.1090/mcom/3014
Received by editor(s): September 9, 2013
Received by editor(s) in revised form: April 24, 2014, and November 4, 2014
Published electronically: August 17, 2015
Additional Notes: The first author was partially supported by NSF Grant DMS-1115331 and by the Minnesota Supercomputing Institute.
The second author was partially supported by NSF Grant DMS-1109325.
The third author was partially supported by NSF Grants DMS-1115331 and DMS-1109325.
Article copyright: © Copyright 2015 American Mathematical Society

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