Daniele Boffi: Finite element approximation of eigenvalue problems in mixed form

Abstract:

The talk starts with a review of the numerical approximation of eigenvalue problems arising from partial differential equations presented in mixed form, such as the Stokes or the Darcy problems. While for standard eigenvalue problems the natural stability conditions for the approximation of the source problem automatically guarantee the convergence of the solutions, for mixed problems the situation is more complex. Extra conditions are requires in addition to the classical inf-sup compatibilities.

We show the implications of the theory to the approximation of the eigenvalue problem associated with the time harmonic Maxwell equations. We present selections of spaces giving optimal convergence, including the popular edge element families, as well as more critical choices where convergence is not achieved or is subject to additional conditions.