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Complex logarithms, Cauchy principal values, and the complex variable boundary element method. (English) Zbl 0811.65102
A derivation of the complex variable boundary element method is presented. This special method is applicable for the approximate solution of two-dimensional mixed boundary value problems for the Laplace equation. The solution is seeked as the real part of a holomorphic function represented by a boundary integral of Cauchy type.
Advantageously is the fact that all integrations for evaluating the discretization matrix can be performed explicitly by use of the complex logarithm. Some care is necessary to choose the right branches in each case. The authors explain in detail the properties of the complex logarithm in context with the jump relations and the Cauchy principal value integrals.
Reviewer: A.Pomp (Stuttgart)

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
31A10 Integral representations, integral operators, integral equations methods in two dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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References:
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