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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Bounds for eigenvalues and condition numbers in the $p$-version of the finite element method
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by Ning Hu, Xian-Zhong Guo and I. Norman Katz PDF
Math. Comp. 67 (1998), 1423-1450 Request permission

Abstract:

In this paper, we present a theory for bounding the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the $p$-version of finite element analysis. Bounds are derived for the eigenvalues and the condition numbers, which are valid for stiffness matrices based on a set of general basis functions that can be used in the $p$-version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the $p$-version, explicit bounds are derived for the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices. We prove that the condition numbers of the stiffness matrices grow like $p^{4(d-1)}$, where $d$ is the number of dimensions. Our results disprove a conjecture of Olsen and Douglas in which the authors assert that “regardless of the choice of basis, the condition numbers grow like $p^{4d}$ or faster". Numerical results are also presented which verify that our theoretical bounds are correct.
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Additional Information
  • Ning Hu
  • Affiliation: Department of Systems Science and Mathematics, Washington University in St. Louis, St. Louis, MO 63130
  • Address at time of publication: Endocardial Solutions, 1350 Energy Lane, St. Paul, MN 55108
  • Email: ning@endo.com
  • Xian-Zhong Guo
  • Affiliation: Department of Mechanical Engineering, Washington University in St. Louis, St. Louis, MO 63130
  • Email: guo@esrd.com
  • I. Norman Katz
  • Affiliation: Department of Systems Science and Mathematics, Washington University in St. Louis, St. Louis, MO 63130
  • Email: katz@zach.wustl.edu
  • Received by editor(s): July 15, 1996
  • Received by editor(s) in revised form: April 1, 1997
  • Additional Notes: This research was supported by Air Force Office of Scientific Research under grant number AFOSR 92-J-0043, and by the National Science Foundation under grant number DMS-9626202. Some of the results presented here are part of the doctoral dissertation of the first author.
  • © Copyright 1998 American Mathematical Society
  • Journal: Math. Comp. 67 (1998), 1423-1450
  • MSC (1991): Primary 65N30; Secondary 65N22, 65F33
  • DOI: https://doi.org/10.1090/S0025-5718-98-00983-1
  • MathSciNet review: 1484898