### Abstract

A variety of problems in materials science and fluid dynamics require the solution of Laplace's equation in multiply connected domains. Integral equation methods are natural candidates for such problems, since they discretize the boundary alone, require no special effort for free boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains in two space dimensions, regardless of shape. Current integral equation methods for the Dirichlet problem, however, require the solution of M independent problems of dimension N, where M is the number of boundary components and N is the total number of points in the discretization. In this paper, we present a new boundary integral equation approach, valid for both interior and exterior problems, which requires the solution of a single linear system of dimension N + M. We solve this system by making use of an iterative method (GMRES) combined with the fast multipole method for the rapid calculation of the necessary matrix vector products. For a two-dimensional system with 200 components and 100 points on each boundary, we gain a speedup of a factor of 100 from the new analytic formulation and a factor of 50 from the fast multipole method. The resulting scheme brings large scale calculations in extremely complex domains within practical reach.

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

Pages (from-to) | 267-278 |

Number of pages | 12 |

Journal | Journal of Computational Physics |

Volume | 105 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1993 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Computer Science Applications

### Cite this

*Journal of Computational Physics*,

*105*(2), 267-278. https://doi.org/10.1006/jcph.1993.1073

**Laplace's Equation and the Dirichlet-Neumann Map in Multiply Connected Domains.** / Greenbaum, A.; Greengard, Leslie; McFadden, G. B.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 105, no. 2, pp. 267-278. https://doi.org/10.1006/jcph.1993.1073

}

TY - JOUR

T1 - Laplace's Equation and the Dirichlet-Neumann Map in Multiply Connected Domains

AU - Greenbaum, A.

AU - Greengard, Leslie

AU - McFadden, G. B.

PY - 1993/4

Y1 - 1993/4

N2 - A variety of problems in materials science and fluid dynamics require the solution of Laplace's equation in multiply connected domains. Integral equation methods are natural candidates for such problems, since they discretize the boundary alone, require no special effort for free boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains in two space dimensions, regardless of shape. Current integral equation methods for the Dirichlet problem, however, require the solution of M independent problems of dimension N, where M is the number of boundary components and N is the total number of points in the discretization. In this paper, we present a new boundary integral equation approach, valid for both interior and exterior problems, which requires the solution of a single linear system of dimension N + M. We solve this system by making use of an iterative method (GMRES) combined with the fast multipole method for the rapid calculation of the necessary matrix vector products. For a two-dimensional system with 200 components and 100 points on each boundary, we gain a speedup of a factor of 100 from the new analytic formulation and a factor of 50 from the fast multipole method. The resulting scheme brings large scale calculations in extremely complex domains within practical reach.

AB - A variety of problems in materials science and fluid dynamics require the solution of Laplace's equation in multiply connected domains. Integral equation methods are natural candidates for such problems, since they discretize the boundary alone, require no special effort for free boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains in two space dimensions, regardless of shape. Current integral equation methods for the Dirichlet problem, however, require the solution of M independent problems of dimension N, where M is the number of boundary components and N is the total number of points in the discretization. In this paper, we present a new boundary integral equation approach, valid for both interior and exterior problems, which requires the solution of a single linear system of dimension N + M. We solve this system by making use of an iterative method (GMRES) combined with the fast multipole method for the rapid calculation of the necessary matrix vector products. For a two-dimensional system with 200 components and 100 points on each boundary, we gain a speedup of a factor of 100 from the new analytic formulation and a factor of 50 from the fast multipole method. The resulting scheme brings large scale calculations in extremely complex domains within practical reach.

UR - http://www.scopus.com/inward/record.url?scp=0001067436&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001067436&partnerID=8YFLogxK

U2 - 10.1006/jcph.1993.1073

DO - 10.1006/jcph.1993.1073

M3 - Article

AN - SCOPUS:0001067436

VL - 105

SP - 267

EP - 278

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -