The Spectral Approximation of Linear Operators with Applications to the Computation of Eigenelements of Differential and Integral Operators

Abstract

We are concerned with the numerical solution of the eigenvalue problem $T\varphi = \lambda \varphi ,\varphi \ne 0$, where T is a linear operator in a Bana,ch space. T may represent a bounded integral operator or a closed differential operator (with bounded inverse). The linear operator T and its approximation $T_n $ are defined in the same space. Perturbation theory is then a suitable framework for our problem. We present in the first part a systematic study of the various notions of convergence $T_n \to T$, which imply the convergence of the eigenvalues with preservation of the multiplicities. The results are then applied to practical methods for integral and differential operators. In the second part, we present convergence rates. Analytic perturbation theory is used to refine on the computed eigenelements of an integral operator, and to produce localization results on the eigenelements. Finally superconvergence results are discussed, both for integral and differential operators.