On variational eigenvalue approximation of semidefinite operators

Eigenvalue problems for semidefinite operators with infinite-dimensional kernels appear, for instance, in electromagnetics. Variational discretizations with edge elements have long been analysed in terms of a discrete compactness property. As an alternative, we show here how the abstract theory can be developed in terms of a geometric property called the vanishing gap (VG) condition. This condition is shown to be equivalent to eigenvalue convergence and intermediate between two different discrete variants of Friedrichs estimates. Next we turn to a more practical means of checking these properties. We introduce the notion of a compatible operator (CO) and show how the previous conditions are equivalent to the existence of such operators with various convergence properties. In particular, the VG condition is shown to be equivalent to the existence of COs satisfying an Aubin–Nitsche estimate. Finally, we give examples demonstrating that the implications not shown to be equivalences, indeed are not.