On the residual inverse iteration for nonlinear eigenvalue problems admitting a Rayleigh functional

The residual inverse iteration is a simple method for solving eigenvalue problems that are nonlinear in the eigenvalue parameter. In this paper, we establish a new expression and a simple bound for the asymptotic convergence factor of this iteration in the special case that the nonlinear eigenvalue problem is Hermitian and admits a so called Rayleigh functional. These results are then applied to discretized nonlinear PDE eigenvalue problems. For this purpose, we introduce an appropriate Hilbert space setting and show the convergence of the smallest eigenvalue of a Galerkin discretization. Under suitable conditions, we obtain a bound for the asymptotic convergence factor that is independent of the discretization. This also implies that the use of multigrid preconditioners yields mesh-independent convergence rates for finite element discretizations of nonlinear PDE eigenvalue problem. A simple numerical example illustrates our findings.