Istituto di Matematica Applicata e Tecnologie Informatiche     
Boffi D., Brezzi F., Gastaldi L. On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Preprint ercim.cnr.ian//1997-1064, 1997.
In the approximation of linear elliptic operators in mixed form, it is well-known that the so-called {it inf-sup} and {it ellipticity in the kernel} property 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 existing literature 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 already quoted commonly used mixed methods) which guarantee optimal error bounds for eigenvalue approximations as well.
Subject Linear elliptic problems, eigenvalues, finite elements

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