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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Bounds for eigenvalues and condition numbers in the $p$-version of the finite element method


Authors: Ning Hu, Xian-Zhong Guo and I. Norman Katz
Journal: Math. Comp. 67 (1998), 1423-1450
MSC (1991): Primary 65N30; Secondary 65N22, 65F33
MathSciNet review: 1484898
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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

DOI: http://dx.doi.org/10.1090/S0025-5718-98-00983-1
PII: S 0025-5718(98)00983-1
Keywords: Eigenvalues; condition number; $p$-version of the finite element method
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.
Article copyright: © Copyright 1998 American Mathematical Society