### Abstract

Capacitance matrix methods provide techniques for extending the use of fast Poisson solvers to arbitrary bounded regions. These techniques are further studied and developed with a focus on the three-dimensional case. A discrete analogue of classical potential theory is used as a guide in the design of rapidly convergent iterative methods. Algorithmic and programming aspects of the methods are also explored in detail. Several conjugate gradient methods are discussed for the solution of the capacitance matrix equation. A fast Poisson solver is developed which is numerically very stable even for indefinite Helmholtz equations. Variants thereof allow substantial savings in primary storage for problems on very fine meshes. Numerical results show that accurate solutions can be obtained at a cost which is proportional to that of the fast Helmholtz solver in use.

Original language | English (US) |
---|---|

Pages (from-to) | 849-879 |

Number of pages | 31 |

Journal | Mathematics of Computation |

Volume | 33 |

Issue number | 147 |

DOIs | |

State | Published - 1979 |

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*Mathematics of Computation*,

*33*(147), 849-879. https://doi.org/10.1090/S0025-5718-1979-0528044-7

**Capacitance matrix methods for the Helmholtz equation on general three dimensional regions.** / O'leary, Dianne P.; Widlund, Olof.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 33, no. 147, pp. 849-879. https://doi.org/10.1090/S0025-5718-1979-0528044-7

}

TY - JOUR

T1 - Capacitance matrix methods for the Helmholtz equation on general three dimensional regions

AU - O'leary, Dianne P.

AU - Widlund, Olof

PY - 1979

Y1 - 1979

N2 - Capacitance matrix methods provide techniques for extending the use of fast Poisson solvers to arbitrary bounded regions. These techniques are further studied and developed with a focus on the three-dimensional case. A discrete analogue of classical potential theory is used as a guide in the design of rapidly convergent iterative methods. Algorithmic and programming aspects of the methods are also explored in detail. Several conjugate gradient methods are discussed for the solution of the capacitance matrix equation. A fast Poisson solver is developed which is numerically very stable even for indefinite Helmholtz equations. Variants thereof allow substantial savings in primary storage for problems on very fine meshes. Numerical results show that accurate solutions can be obtained at a cost which is proportional to that of the fast Helmholtz solver in use.

AB - Capacitance matrix methods provide techniques for extending the use of fast Poisson solvers to arbitrary bounded regions. These techniques are further studied and developed with a focus on the three-dimensional case. A discrete analogue of classical potential theory is used as a guide in the design of rapidly convergent iterative methods. Algorithmic and programming aspects of the methods are also explored in detail. Several conjugate gradient methods are discussed for the solution of the capacitance matrix equation. A fast Poisson solver is developed which is numerically very stable even for indefinite Helmholtz equations. Variants thereof allow substantial savings in primary storage for problems on very fine meshes. Numerical results show that accurate solutions can be obtained at a cost which is proportional to that of the fast Helmholtz solver in use.

UR - http://www.scopus.com/inward/record.url?scp=0001390840&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001390840&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1979-0528044-7

DO - 10.1090/S0025-5718-1979-0528044-7

M3 - Article

VL - 33

SP - 849

EP - 879

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 147

ER -