This paper sums up some recent studies related to the numerical solution of boundary value problems deriving from Maxwell's equations. These studies bring to light the theoretical origins of the ‘corner paradox’ pointed out by numerical experiments for years: In a domain surrounded by a perfect conductor, a ‘nodal’ discretization can approximate the electromagnetic field only if the domain has no reentrant corners or edges. The explanation lies in a mathematical curiosity: two different interpretations of the same variational equation, which are both well-posed and lead either to the physical or a spurious solution! Two strategies which were recently proposed to remedy this flaw of nodal elements are described.