Improved spectral multigrid methods for periodic elliptic problems.

*(English)*Zbl 0569.65084The authors introduce for the periodic case some refinements which can substantially improve the overall efficiency of the spectral multigrid method. It is shown that by evaluating the fluxes at the midpoints between the usual collocation points a ”midpoint” pseudospectral discretization is obtained which eliminates the need for filtering the highest modes, resulting in improved accuracy and efficiency. For the isotropic case a weighted residual relaxation scheme with a greatly improved smoothing rate is introduced and it is shown that by property scaling the relaxation parameters, the smoothing rates obtained for constant coefficient problems also hold even when the coefficients are not constant. Some comparisons of choices for coarse grid operators and residual transfers are also made. The authors introduce another relaxation scheme, based on defect corrections, which is efficient even for highly anisotropic problems. Numerical results are given.

Reviewer: K.Najzar

##### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

35B10 | Periodic solutions to PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

##### Keywords:

midpoint pseudospectral discretization; refinements; spectral multigrid method; collocation; weighted residual relaxation scheme; smoothing rate; comparisons; coarse grid operators; defect corrections; Numerical results
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##### References:

[1] | Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods, () · Zbl 0561.76076 |

[2] | Orszag, S.A., J. comput. phys., 37, 70, (1980) |

[3] | Wong, Y.S.; Zang, T.A.; Hussaini, M.Y., Efficient iterative techniques for the solution of spectral equations, () |

[4] | Zang, T.A.; Wong, Y.S.; Hussaini, M.Y., J. comput. phys., 48, 485, (1982) |

[5] | {\scT. A. Zang, Y. S. Wong, AND M. Y. Hussaini}, “Spectral Multigrid Methods for Elliptic Equations II,” J. Comput. Phys., in press. · Zbl 0543.65071 |

[6] | Brandt, A., Math. comput., 31, 333, (1977) |

[7] | Brandt, A., () |

[8] | Kaufman, E.H.; Taylor, G.D., Internat. J. numer. methods engrg., 9, 297, (1975) |

[9] | McCrory, R.L.; Orszag, S.A., J. comput. phys., 37, 93, (1980) |

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