Quadratic projection methods for approximating the spectrum of self-adjoint operators

The pollution-free approximation of the spectrum for self-adjoint operators using a quadratic projection method has recently been studied. Higher-order pollution-free approximation can be achieved by combining this technique with a method due to Kato. To illustrate, an example from magnetohydrodynamics is considered. Whether or not this procedure converges to the whole spectrum is unknown. Combining the quadratic method with the Galerkin method, we derive procedures that do converge to the whole spectrum and without pollution.