  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 can be written as the eigenvalue problem of
some holomophic Fredholm operator function. Focusing on the finite element
methods (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 compact convergence
of the discrete operators in some 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 various finite element methods for
eigenvalue problems such as the Dirhcilet eigenvalue problem and the biharmonic
eigenvalue problem.
