On the problem of spurious eigenvalues

in the approximation of linear elliptic problems

in mixed form

Authors:
Daniele Boffi, Franco Brezzi and Lucia Gastaldi

Journal:
Math. Comp. **69** (2000), 121-140

MSC (1991):
Primary 65N30; Secondary 65N25

DOI:
https://doi.org/10.1090/S0025-5718-99-01072-8

Published electronically:
February 19, 1999

MathSciNet review:
1642801

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Abstract | References | Similar Articles | Additional Information

Abstract: In the approximation of linear elliptic operators in mixed form, it is well known that the so-called *inf-sup* and *ellipticity in the kernel* properties are sufficient (and, in a sense to be made precise, necessary) in order to have good approximation properties and optimal error bounds. One might think, in the spirit of Mercier-Osborn-Rappaz-Raviart and in consideration of the good behavior of commonly used mixed elements (like Raviart-Thomas or Brezzi-Douglas-Marini elements), that these conditions are also sufficient to ensure good convergence properties for eigenvalues. In this paper we show that this is not the case. In particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues. Such bad behavior is proved analytically and demonstrated in numerical experiments. We also present additional assumptions (fulfilled by the commonly used mixed methods already mentioned) which guarantee optimal error bounds for eigenvalue approximations as well.

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Additional Information

**Daniele Boffi**

Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, 27100 Pavia, Italy

Email:
boffi@ian.pv.cnr.it

**Franco Brezzi**

Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia and Istituto di Analisi Numerica del C.N.R., 27100 Pavia, Italy

Email:
brezzi@ian.pv.cnr.it

**Lucia Gastaldi**

Affiliation:
Dipartimento di Matematica, Università di Roma “La Sapienza”, 00185 Roma, Italy

Email:
gastaldi@ian.pv.cnr.it

DOI:
https://doi.org/10.1090/S0025-5718-99-01072-8

Keywords:
Mixed finite element methods,
spurious eigenvalues

Received by editor(s):
July 8, 1997

Received by editor(s) in revised form:
March 17, 1998

Published electronically:
February 19, 1999

Additional Notes:
Partially supported by I.A.N.-C.N.R. Pavia, by C.N.R. under contracts no. 95.01060.12, 96.03853.CT01, 97.00892.CT01, and by MURST

Article copyright:
© Copyright 1999
American Mathematical Society