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Mathematics of Computation

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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
Published electronically: February 19, 1999
MathSciNet review: 1642801
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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

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

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

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