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

A. Greenbaum, Leslie Greengard, G. B. McFadden

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)267-278
Number of pages12
JournalJournal of Computational Physics
Volume105
Issue number2
DOIs
StatePublished - Apr 1993

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Laplace equation
Integral equations
integral equations
Boundary integral equations
Materials science
Fluid dynamics
Iterative methods
multipoles
Linear systems
Dirichlet problem
free boundaries
fluid dynamics
materials science
linear systems
formulations
products
matrices

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Computer Science Applications

Cite this

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

In: Journal of Computational Physics, Vol. 105, No. 2, 04.1993, p. 267-278.

Research output: Contribution to journalArticle

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