New A Priori FEM Error Estimates for Eigenvalues

We analyze the Ritz--Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, what is apparently the first truly a priori error estimates that show the levels of the eigenvalue errors depending on approximability of eigenfunctions in the corresponding eigenspace. These estimates reflect a known phenomenon that different eigenfunctions in the corresponding eigenspace may have different approximabilities, thus resulting in different levels of errors for the approximate eigenvalues. For clustered eigenvalues, we derive eigenvalue error bounds that do not depend on the width of the cluster. Our results are readily applicable to the classical Ritz method for compact symmetric integral operators and to finite element method eigenvalue approximation for symmetric positive definite differential operators.