Compatible Discretizations for Eigenvalue Problems

Abstract

The choice of discrete spaces for a variationally posed symmetric and compact eigenvalue problem corresponding a source problem is discussed. Any standard Galerkin discretization space that is convergent for the source problem automatically performs well for the eigenvalue problem. On the other hand, mixed discretizations that are convergent (satisfying the classical Brezzi conditions) exhibit spurious low frequency eigenmodes. Examples of discretizations with spurious modes are presented. Moreover, necessary and sufficient conditions on mixed discretization are established for the (ordered) discrete eigenvalues to converge to the corresponding continuous eigenvalues. The theory is applied to the determination of band gaps for photonic crystals and evolution problems.