This article discusses finite element Galerkin schemes for a number of linear
model problems in electromagnetism. The finite element schemes are introduced
as discrete differential forms, matching the coordinate-independent
statement of Maxwell's equations in the calculus of differential forms. The
asymptotic convergence of discrete solutions is investigated theoretically. As
discrete differential forms represent a genuine generalization of conventional
Lagrangian finite elements, the analysis is based upon a judicious adaptation
of established techniques in the theory of finite elements. Risks and difficulties
haunting finite element schemes that do not fit the framework of discrete differential
forms are highlighted.