Refined estimates for the errors in eigenvalue and eigenvector approximation by finite element methods are derived. More generally, Galerkin methods for selfadjoint problems are concerned. A special look is casted on the important case of multiple eigenvalues and eigenvectors.
First two lemmas are derived on two functions (one in the Hilbert space and the other in a corresponding subspace) of nonnegative real variable, as a performer of the eigenvalues, and a third lemma on an error estimation. The main part of the paper defines and verifies in a very sophisticated manner an estimative criterion for the eigenvalue and eigenvector errors of the Galerkin method, in terms of some approximability quantities. Then, numerical computations for a finite element approximation of a problem with double eigenvalues are presented, each double eigenvalue having associated eigenvectors of strikingly different approximation properties. So, preasymptotic and asymptotic behaviour of the errors on number of intervals is separated.