Boundary Integral Equations

  • Kendall Atkinson
  • Weimin Han
Part of the Texts in Applied Mathematics book series (TAM, volume 39)


In Chapter 9, we examined finite element methods for the numerical solution of Laplace’s equation. In this chapter, we propose an alternative approach. We introduce the idea of reformulating Laplace’s equation as a boundary integral equation (BIE), and then we consider the numerical solution of Laplace’s equation by numerically solving its reformulation as a BIE. Some of the most important boundary value problems for elliptic partial differential equations have been studied and solved numerically by this means; and depending on the requirements of the problem, the use of BIE reformulations may be the most efficient means of solving these problems. Examples of other equations solved by use of BIE reformulations are the Helmholtz equation (Δu + λu = 0) and the biharmonic equation (Δ2u = 0). We consider here the use of boundary integral equations in solving only planar problems for Laplace’s equation. For the domain D for the equation, we restrict it or its complement to be a simply connected set with a smooth boundary S. Most of the results and methods given here will generalize to other equations (e.g., Helmholtz’s equation).


Dirichlet Problem Neumann Problem Boundary Integral Equation Double Layer Potential Exterior Problem 
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Copyright information

© Springer-Verlag New York, Inc. 2001

Authors and Affiliations

  • Kendall Atkinson
    • 1
  • Weimin Han
    • 2
    • 3
  1. 1.Department of Mathematics, Department of Computer ScienceUniversity of IowaIowa CityUSA
  2. 2.Department of MathematicsUniversity of IowaIowa CityUSA
  3. 3.Department of MathematicsZhejiang UniversityHangzhouPeople’s Republic of China

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