Improved error estimates for the perturbed Galerkin method applied to a class of generalized eigenvalue problems

Abstract

We consider the generalized eigenvalue problem x- Kx = μBx in a complex Banach space E. Here, K and B are bounded linear operators, B is compact, and 1 is not in the spectrum of K. If { E n : n = 1, 2,…} is a sequence of closed subspaces of E and P n : E → E n is a linear projection which maps E onto E n , then we consider the sequence of approximate eigenvalue problems { x n - P n Kx n = μP n Bx n in E n : n = 1, 2,…}. Assuming that ∥ K- P n K∥ → 0 and t| B- P n B∥ → 0 as n → ∞, we prove the convergence of sequences of eigenvalues and eigenelements of the approximate eigenvalue problem to eigenvalues and eigenelements of the original eigenvalue problem, and establish upper bounds for the errors. These error bounds are sharper than those given by Vainikko in Ref. 2 for the more general problem x = μTx in E, T linear and compact, and the sequence of approximate problems { x n = μT n x n in E n : n = 1, 2,…}, and do not involve the operator S n = T n - P n T ∥; E n .