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**Separation-of-variables as a preconditioner for an iterative Helmholtz solver.**
*(English)*
Zbl 1013.65117

Summary: A preconditioned iterative method based on separation-of-variables for solving the Helmholtz equation in an inhomogeneous medium is tested. The preconditioner is constructed by approximating the wavenumber by a sum of two terms, one depending only on one spatial coordinate, say \(x\), and the other depending on the remaining coordinates. The Helmholtz equation can be solved efficiently if the wavenumber has such a separable form.

First, an eigenvalue-eigenvector decomposition is applied in the \(x\)-direction. With these eigenvectors, a change of variables is performed in order to obtain a set of independent systems with one dimension less than the original one. For smooth models and low frequencies, the convergence rate with this preconditioner is satisfactory. Unfortunately, it rapidly deteriorates when the roughness of the model or the frequency increases.

Examples from seismic modeling are given to illustrate this behaviour. Moreover, numerical evidence is presented that suggests that the decomposition of the wavenumber in the sum of two terms cannot be improved with this approach.

First, an eigenvalue-eigenvector decomposition is applied in the \(x\)-direction. With these eigenvectors, a change of variables is performed in order to obtain a set of independent systems with one dimension less than the original one. For smooth models and low frequencies, the convergence rate with this preconditioner is satisfactory. Unfortunately, it rapidly deteriorates when the roughness of the model or the frequency increases.

Examples from seismic modeling are given to illustrate this behaviour. Moreover, numerical evidence is presented that suggests that the decomposition of the wavenumber in the sum of two terms cannot be improved with this approach.

### MSC:

65N06 | Finite difference methods for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

86A15 | Seismology (including tsunami modeling), earthquakes |

### Keywords:

Helmholtz equation; preconditioning; separation of variables; iterative solver; numerical examples; finite difference method; iterative method; wavenumber; eigenvalue-eigenvector decomposition; convergence; seismic modeling### Software:

symrcm
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\textit{R. E. Plessix} and \textit{W. A. Mulder}, Appl. Numer. Math. 44, No. 3, 385--400 (2003; Zbl 1013.65117)

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### References:

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