POSTERIORI ERROR BOUNDS FOR EIGEN-ELEMENTS OF LINEAR-OPERATORS

The theoretical framework of this study is presented in Sect. 1, with a review of practical numerical methods. The linear operator T and its approximation Tn are defined in the same Banach space, which is a very common situation. The notion of strong stability for Tn is essential and cannot be weakened without introducing a numerical instability [2]. If T (or its inverse) is compact, most numerical methods are strongly stable. Without compactness for T(T-1) they may not be strongly stable [20]. In Sect. 2 we establish error bounds valid in the general setting of a strongly stable approximation of a closed T. This is a generalization of Vainikko [24, 25] (compact approximation). Osborn [19] (uniform and collectivity compact approximation) and Chatelin and Lemordant [6] (strong approximation), based on the equivalence between the eigenvalues convergence with preservation of multiplicities and the collectively compact convergence of spectral projections. It can be summarized in the following way: λ, eigenvalue of T of multiplicity m is approximated by m numbers, λn is their arithmetic mean. λ-λn and the gap between invariant subspaces are of order εn={norm of matrix}(T-Tn){norm of matrix}P. If Tn * converges to T*, pointwise in X*, the principal term in the error on {divides}λ-λn{divides} is {Mathematical expression}. And for projection methods, with Tn=πnT, we get the bound {Mathematical expression}. It applies to the finite element method for a differential operator with a noncompact resolvent. A posteriori error bounds are given, and the generalized Rayleigh quotient {Mathematical expression}TPn appears to be an approximation of λ of the second order, as in the selfadjoint case [12]. In Sect. 3, these results are applied to the Galerkin method and its Sloan variant [22], and to approximate quadrature methods. The error bounds and the generalized Rayleigh quotient are numerically tested in Sect. 4. © 1979 Springer-Verlag.