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A Survey of Boundary Integral Equation Methods for the Numerical Solution of Laplace’s Equation in Three Dimensions

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Numerical Solution of Integral Equations

Part of the book series: Mathematical Concepts and Methods in Science and Engineering ((MCSENG,volume 42))

Abstract

The principal reformulations of Laplace’s equation as boundary integral equations (BIEs) are described, together with results on their solvability and the regularity of their solutions. The numerical methods for solving BIEs are categorized, based on whether the method uses local or global approximating functions, whether the method is of collocation or Galerkin type, and whether the equation being solved is defined on a region whose boundary is smooth or only piecewise smooth. Some of the major ideas in the mathematical analysis of these numerical methods are outlined. Certain problems are associated with all numerical methods for boundary integral equations. Principal among these are numerical integration and the iterative solution of linear systems of equations. The research literature for these topics as they arise in solving BIEs is reviewed, and some of the major ideas are discussed.

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  1. Department of Mathematics, University of Iowa, Iowa City, Iowa, 52242, USA

    K. E. Atkinson

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  1. K. E. Atkinson
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Editors and Affiliations

  1. Department of Mathematical Sciences, University of Nevada, Las Vegas, Nevada, 89154, USA

    Michael A. Golberg

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Atkinson, K.E. (1990). A Survey of Boundary Integral Equation Methods for the Numerical Solution of Laplace’s Equation in Three Dimensions. In: Golberg, M.A. (eds) Numerical Solution of Integral Equations. Mathematical Concepts and Methods in Science and Engineering, vol 42. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-2593-0_1

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  • DOI: https://doi.org/10.1007/978-1-4899-2593-0_1

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