Regular Convergence and Finite Element Methods for Eigenvalue Problems

Regular convergence, together with various other types of convergence, has been studied since the 1970s for the discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of compact operators whose spectral problem is equivalent to the eigenvalue problem of some holomophic Fredholm operator function. Focusing on the finite element methods (conforming, non-conforming, discontinuous Galerkin, etc.), we show that the regular convergence of discrete holomorphic operator functions follows from the approximation property of the finite element spaces and the point convergence of the discrete operators in a suitable Sobolev space. The convergence for eigenvalues is then obtained using the discrete approximation theory for the eigenvalue problems of holomorphic Fredholm operator functions. The result can be used to show the convergence of many finite element methods for eigenvalue problems such as the Dirhcilet eigenvalue problem and the biharmonic eigenvalue problem.