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

We propose an extension to the double iteration method, the Multipower Defect Correction method, which is used to refine approximate eigenelements of integral operators. It consists of an inner/outer iteration where, inside a defect correction iteration, p steps of a power iteration are used. The approximate inverse used in the defect correction is built with an approximation to the reduced resolvent operator of a small size discretization of the integral operator. The kernel of the integral operator may be weakly singular. The proof of the convergence of this Multipower Defect Correction method is presented and a numerical example illustrates the behavior of the method. AMS: 45P05, 33F05, 32A55, 45C05, 34L16, 34L05.