Sur les bornes d'erreur a posteriori pour les �l�ments propres d'op�rateurs lin�aires

Abstract

The theoretical framework of this study is presented in Sect. 1, with a review of practical numerical methods. The linear operatorT and its approximationT n are defined in the same Banach space, which is a very common situation. The notion of strong stability forT n is essential and cannot be weakened without introducing a numerical instability [2]. IfT (or its inverse) is compact, most numerical methods are strongly stable. Without compactness forT(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 closedT. 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: ?, eigenvalue ofT of multiplicitym is approximated bym numbers,? n is their arithmetic mean.?-? n and the gap between invariant subspaces are of order? n =?(T-T n)?P. IfT n * converges toT *, pointwise inX *, the principal term in the error on ??-? n ? is $$\frac{1}{m}|tr (T - T_n )P|$$ . And for projection methods, withT n=? n T, we get the bound $$|tr (T - T_n )P| \leqq C ||(1 - \pi _n )P|| ||(1 - \pi _n^* )P*||$$ . It applies to the finite element method for a differential operator with a noncompact resolvent. Aposteriori error bounds are given, and thegeneralized Rayleigh quotient $$\frac{1}{m}tr TP_n $$ TP n appears to be an approximation of ? 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.