## Abstract

The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N^{2} log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.

Original language | English |
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Pages (from-to) | 91-128 |

Number of pages | 38 |

Journal | Journal of Scientific Computing |

Volume | 21 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2004 |

Externally published | Yes |

### Bibliographical note

Funding Information:The research of E.B. was partially supported by the University Research Grant of the University of Calgary. The research of M.I. was partially supported by the VPR fund for promotion of research at the Technion.

## Keywords

- Equations in complex geometries
- Fast spectral direct solver
- Preconditioned iterative algorithm for elliptic equations
- The Poisson equation
- The modified helmholtz equation