Convergence of Multipower Defect-Correction for spectral computations of integral operators

In this paper we propose the Multipower Defect-Correction method, a generalization of the double iteration, to compute a cluster of eigenvalues and the associated invariant subspace from discretized integral operators. It consists of an inner/outer iteration where, inside a defect-correction iteration, p power iteration steps are performed. The approximate inverse used in the defect correction is built with an approximation to the reduced resolvent operator of a coarse discretization of the integral operator. The proposed method computes eigenpairs approximations by refining initial approximations obtained from a coarser dimensional problem. It is therefore meant for large dimensional problems. Furthermore, the kernel of the integral operator may be weakly singular. We provide a proof for the convergence of this Multipower Defect-Correction method. A numerical example illustrating the theory and the behavior of the method is also presented.