Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number µ

Resumen

We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number μ" role="presentation" style="position: relative;">??µ µ , approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a τ" role="presentation" style="position: relative;">??t t preconditioning when the variable coefficient wave number μ" role="presentation" style="position: relative;">??µ µ is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented.