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**Fast and high accuracy multigrid solution of the three dimensional Poisson equation.**
*(English)*
Zbl 0927.65141

Multigrid methods for solving the Poisson equation in three-dimensional domains are studied. For the discretization a fourth-order compact difference scheme (19-point scheme) and the usual second-order central difference scheme (7-point scheme) are considered.

Using a Fourier analysis it is shown that the smoothing factor of the lexicographic Gauss-Seidel method is smaller in the case of the 19-point scheme than in the case of the 7-point scheme. Furthermore, a new method of Fourier smoothing analysis to study a partially decoupled red-black Gauss-Seidel relaxation with the 19-point scheme is proposed. This analysis shows that the smoothing factor of the red-black Gauss-Seidel smoother is smaller than that of the lexicographic Gauss-Seidel method. The numerical experiments presented confirm these theoretical results.

The influence of several grid transfer operators on the convergence behaviour and the efficiency of the multigrid method is studied by numerical experiments.

Using a Fourier analysis it is shown that the smoothing factor of the lexicographic Gauss-Seidel method is smaller in the case of the 19-point scheme than in the case of the 7-point scheme. Furthermore, a new method of Fourier smoothing analysis to study a partially decoupled red-black Gauss-Seidel relaxation with the 19-point scheme is proposed. This analysis shows that the smoothing factor of the red-black Gauss-Seidel smoother is smaller than that of the lexicographic Gauss-Seidel method. The numerical experiments presented confirm these theoretical results.

The influence of several grid transfer operators on the convergence behaviour and the efficiency of the multigrid method is studied by numerical experiments.

Reviewer: M.Jung (Chemnitz)

### MSC:

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

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 |

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

### Keywords:

Poisson equation; finite difference method; multigrid method; Fourier analysis; smoothing properties; numerical examples; lexicographic Gauss-Seidel method; red-black Gauss-Seidel relaxation; convergence; three-dimensional domains### Software:

Wesseling### References:

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