Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix. (English) Zbl 0972.65031

The authors study a new class of preconditioning methods to solve large and sparse linear systems with highly indefinite complex-symmetric systems. The starting observation is that the Krylov subspace method sometimes degrades when incomplete Cholesky factorizations are used as a preconditioner. The main underlying reason for this is that the eigenvalues with negative real part, on their move towards 1 have to cross the origin. This can lead to a clustering of eigenvalues near zero. The proposed remedy is to slightly move the spectrum of the original system matrix along the imaginary axis.
The paper contains a theoretical analysis and extensive numerical tests in the case of the Helmholtz equation in several domains. The superiority of their method is clearly visible in these cases.


65F35 Numerical computation of matrix norms, conditioning, scaling
65F10 Iterative numerical methods for linear systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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