Complex logarithms, Cauchy principal values, and the complex variable boundary element method.

*(English)*Zbl 0811.65102A 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.

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 |

##### Keywords:

complex variable boundary element method; Laplace equation; boundary integral of Cauchy type; complex logarithm
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\textit{R. J. Whitley} and \textit{T. V. Hromadka II}, Appl. Math. Modelling 18, No. 8, 423--428 (1994; Zbl 0811.65102)

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##### References:

[1] | Hromadka, T.; Lai, C., The complex variable boundary element method in engineering analysis, (1987), Springer-Verlag New York · Zbl 0609.65078 |

[2] | Hromadka II, T. and Whitley, R. Expansion of the CVBEM into a Series Using Fractals, submitted for publication · Zbl 1182.76935 |

[3] | Conway, J., Functions of one complex variable, (1978), Springer-Verlag New York |

[4] | Henrici, P., Applied and computational complex analysis III, (1986), Wiley-Interscience New York |

[5] | Muskhelishvili, N., Singular integral equations, (1991), Dover New York, reprint · Zbl 0108.29203 |

[6] | Palka, B., An introduction to complex function theory, (1990), Springer-Verlag New York |

[7] | Gakov, F., Boundary value problems, (1990), Dover New York, reprint |

[8] | Whitley, R. and Hromadka II, T. Numerical solution of the Dirichlet problem via a density theorem, Numer. Method PDE, forthcoming · Zbl 1117.65313 |

[9] | Hromadka, T.; Whitley, R., Error bounds for numerical solutions of partial differential equations, Numer. methods PDE, 7, 339-346, (1991) · Zbl 0753.65084 |

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