Abstract :
We discuss the finite element approximation of eigenvalue problems associated
with compact operators. While the main emphasis is on symmetric problems,
some comments are present for non-self-adjoint operators as well. The topics
covered include standard Galerkin approximations, non-conforming approximations,
and approximation of eigenvalue problems in mixed form. Some
applications of the theory are presented and, in particular, the approximation
of the Maxwell eigenvalue problem is discussed in detail. The final part
tries to introduce the reader to the fascinating setting of differential forms and
homological techniques with the description of the Hodge–Laplace eigenvalue
problem and its mixed equivalent formulations. Several examples and numerical
computations complete the paper, ranging from very basic exercises to
more significant applications of the developed theory.
