REGULAR CONVERGENCE AND FINITE ELEMENT METHODS FOR EIGENVALUE PROBLEMS

Regular convergence, together with other types of convergence, have been studied since the 1970s for discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of a compact operator T that can be written as an eigenvalue problem of a holomorphic Fredholm operator function F(& eta;) = T- & eta; 1 I. Focusing on finite element methods (conforming, discontinuous Galerkin, non-conforming, etc.), we show that the regular convergence of the discrete holomorphic operator functions F,,, to F follows from the compact convergence of the discrete operators T,,, to T. The convergence of the eigenvalues is then obtained using abstract approximation theory for the eigenvalue problems of holomorphic Fredholm operator functions. The result can be used to prove the convergence of various finite element methods for eigenvalue problems such as the Dirichlet eigenvalue problem and the biharmonic eigenvalue problem.