Michael Strauss

Lyonell Boulton, Michael Strauss

Let A be a self-adjoint operator acting on a Hilbert space. The notion of second order spectrum of A relative to a given finite-dimensional subspace L has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. We establish in this paper a general framework allowing us to determine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of A. Our theoretical findings are supported by various numerical experiments on the computation of inclusions for eigenvalues of benchmark differential operators via finite element bases.

Lyonell Boulton, Michael Strauss

We discuss how to compute certified enclosures for the eigenvalues of benchmark linear magnetohydrodynamics operators in the plane slab and cylindrical pinch configurations. For the plane slab, our method relies upon the formulation of an eigenvalue problem associated to the Schur complement, leading to highly accurate upper bounds for the eigenvalue. For the cylindrical configuration, a direct application of this formulation is possible, however, it cannot be rigourously justified. Therefore in this case we rely on a specialized technique based on a method proposed by Zimmermann and Mertins. In turns this technique is also applicable for finding accurate complementary bounds in the case of the plane slab. We establish convergence rates for both approaches.