Incomplete block factorization preconditioning for linear systems arising in the numerical solution of the Helmholtz equation. (English) Zbl 0854.65086

The paper is concerned with the numerical solution of the Dirichlet boundary value problem in a plane domain for the inhomogeneous Helmholtz equation with a complex coefficient. A five-point finite difference discretization yields a linear system with a complex symmetric coefficient matrix, being block-tridiagonal and also being a complex perturbation of an \(M\)-matrix. It is to be solved by conjugate gradient type methods, here by the biconjugate gradient method, see e.g. D. A. H. Jacobs [IMA J. Numer. Anal. 6, 447-452 (1986; Zbl 0614.65028)].
Hereby the problem of choosing an effective preconditioner remains. The suggestion to use preconditioners found to be good for the unperturbed block-tridiagonal matrix also for the perturbed matrices tends to be unsatisfactory when the perturbation is relatively large. For this case, two incomplete block factorizations for the complex system matrix and for its real part are established (in the case of small mesh size). Numerical test results (Dirichlet problem in the unit square) show that the use of these factorizations as preconditioners gives considerably better convergence results than the use of preconditioners used earlier.


65N06 Finite difference methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F35 Numerical computation of matrix norms, conditioning, scaling


Zbl 0614.65028
Full Text: DOI


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