On spectral approximation. Part 1. The problem of convergence

On spectral approximation. Part 2. Error estimates for the Galerkin method

“…Lemma 1 of section 2 of [3] shows that PI implies the inequality sup || R z (A h ) x || rg c, VzeF for h small enough, c independent of h. We note that we shall not use any more PI explicitely. Consequently, in the proofs of lemma 3 and theorem 1, the statements for the adjoints operators are obtained in the same way as for the direct operators.…”

“…In order to illustrate this theorem, we consider the example developped in section 4 of part 1 of this paper [3] ; one can prove by Rappaz' method of élimination used in [6] the existence of an infinité number of isolated eigenvalues of finite multiplicities; by supposing the coefficients a, P, ... sufficiently smooth, one vérifies that the corresponding eigensubspaces are subsets of H 2 x (H 1 ) 2 ; consequently y h = O (h), y* = O (h) and the estimâtes of theorem 3 a, b are of order h 2 .…”

“…In part 1 of this paper [3], we have been concernée! with the problem of convergence in spectral approximation; since the theory we have developped has received concrete applications for non compact operators only in connection with the Galerkin method, we shall now restrict ourself to this case.…”

Accelerated spectral approximation

“…In [11] Vainikko obtained similar results under (discrete) regular approximation. Subsequently, numerical solutions of eigenvalue problems for compact as well as noncompact operators have been studied extensively ( [1], [3], [4], [6], [7], [9]). …”

On the convergence of the finite element approximation of eigenfrequencies and eigenvectors to Maxwell's boundary value problem

“…to the eigenvalue approximation of boundary value problems arising in [11]. For other general treatments of eigenvalue approximation we refer to [2], [3], [4], [13] and [20].…”