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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Convergence and optimality of adaptive mixed finite element methods
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by Long Chen, Michael Holst and Jinchao Xu PDF
Math. Comp. 78 (2009), 35-53 Request permission


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
  • MR Author ID: 735779
  • Email:
  • Michael Holst
  • Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093
  • MR Author ID: 358602
  • Email:
  • Jinchao Xu
  • Affiliation: The School of Mathematical Science, Peking University, and Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16801
  • MR Author ID: 228866
  • Email:
  • 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.
  • © Copyright 2008 American Mathematical Society
  • Journal: Math. Comp. 78 (2009), 35-53
  • MSC (2000): Primary 65N12, 65N15, 65N30, 65N50, 65Y20
  • DOI:
  • MathSciNet review: 2448696