Bounds for eigenvalues and condition numbers in the -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

DOI:
https://doi.org/10.1090/S0025-5718-98-00983-1

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 -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 -version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the -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 , where 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 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:
https://doi.org/10.1090/S0025-5718-98-00983-1

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