Speaker
Description
In this talk we will present eigenvalue estimates for the solutions of operator Lyapunov equations with a noncompact (but relatively Hilbert Schmidt) control operator. We compute eigenvalue estimates from Galerkin discretizations of Lyapunov equations and discuss the appearance of spurious (non convergent) discrete eigenvalues. This phenomenon is called the spectral pollution. Our main tools, which are of independent interest in their own right, are new improved asymptotic estimates for the eigenvalue decay in the case of control operators of large or infinite rank as well as a rank criterion for determining the part of the spectrum of the discrete Gramian which is converging to the eigenvalues of the full Gramian (pollution free part of the discrete spectrum). We test our theoretical results on a collection of academic prototypes using both finite element as well as spectral element discretizations. We will also give a general overview of the regularity theory for the eigenvectors of solutions of Lyapunov operator equation and its influence on construction high order piecewise polynomial approximations ($hp$-adaptive finite elements).
