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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 prop- erties are sucient (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 sucient 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 nite element approximations that satisfy the above properties but exhibit spurious eigenvalues. Such bad behavior is proved analytically and demonstrated in nu- merical experiments. We also present additional assumptions (fullled by the commonly used mixed methods already mentioned) which guarantee optimal error bounds for eigenvalue approximations as well.

On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form