Convergence of multipower defect-correction for spectral computations of integral operators. (English) Zbl 1293.65176
Summary: 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.
MSC:
65R20 | Numerical methods for integral equations |
45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
45P05 | Integral operators |
45C05 | Eigenvalue problems for integral equations |
47G10 | Integral operators |
Keywords:
integral operator; projection approximation; eigenvalue approximation; weakly singular kernel; multipower defect-correction method; double iteration, cluster of eigenvalues; invariant subspace; inner/outer iteration; resolvent operator; eigenpair; convergence; numerical exampleReferences:
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