Sur les bornes d'erreur a posteriori pour les éléments propres d'opérateurs linéaires
Tóm tắt
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: λ, eigenvalue ofT of multiplicitym is approximated bym numbers,λ
n
is their arithmetic mean.λ-λ
n
and the gap between invariant subspaces are of orderε
n
=‖(T-T
n)‖P. IfT
n
*
converges toT
*, pointwise inX
*, the principal term in the error on ∣λ-λ
n
∣ is
$$\frac{1}{m}|tr (T - T_n )P|$$
. And for projection methods, withT
n=π
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 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.
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