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Convergence and optimality of adaptive mixed finite element methods


Authors: Long Chen, Michael Holst and Jinchao Xu
Journal: Math. Comp. 78 (2009), 35-53
MSC (2000): Primary 65N12, 65N15, 65N30, 65N50, 65Y20
DOI: https://doi.org/10.1090/S0025-5718-08-02155-8
Published electronically: June 30, 2008
MathSciNet review: 2448696
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Abstract: The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasi-orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.


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Additional Information

Long Chen
Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697
Email: chenlong@math.uci.edu

Michael Holst
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093
Email: mholst@math.ucsd.edu

Jinchao Xu
Affiliation: The School of Mathematical Science, Peking University, and Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16801
Email: xu@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02155-8
Received by editor(s): April 3, 2006
Received by editor(s) in revised form: November 21, 2007
Published electronically: June 30, 2008
Additional Notes: The first two authors were supported in part by NSF Awards 0411723 and 022560, in part by DOE Awards DE-FG02-04ER25620 and DE-FG02-05ER25707, and in part by NIH Award P41RR08605.
The third author was supported in part by NSF DMS-0619587, DMS-0609727, NSFC-10528102 and the Alexander Humboldt foundation.
Article copyright: © Copyright 2008 American Mathematical Society

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