Spectral pollution and eigenvalue bounds

The Galerkin method can fail dramatically when applied to eigenvalues in gaps of the extended essential spectrum. This phenomenon, called spectral pollution, is notoriously difficult to predict and it can occur in models from relativistic quantum mechanics, solid state physics, magnetohydrodynamics and elasticity theory. The purpose of this survey paper is two-fold. On the one hand, it describes a rigorous mathematical framework for spectral pollution. On the other hand, it gives an account on two complementary state-of-the-art Galerkin-type methods for eigenvalue computation which prevent spectral pollution completely.