Evaluation of a boundary integral representation for the conformal mapping of the unit disk onto a simply-connected domain
The conformal mapping \(g\) of the interior or exterior of the unit circle onto a simply-connected domain \(\Omega\) with corners is represented as a boundary integral containing \(f|_{\partial\Omega}\). Hereby, the inverse mapping \(f:= g^{- 1}\), which can be obtained e.g. as solution of Symm's integral equation, is considered as known on the boundary \(\partial\Omega\). Due to the corners of \(\Omega\) the representation for \(g\) contains integrals over the product of weakly singular functions and is ill- conditioning near the corner points. This leads to significant numerical problems. The authors describe and analyse quadrature schemes appropriate for these difficulties based on product integration. The efficacy of this method is illustrated with numerical experiments.
scientific article; zbMATH DE number 803466
scientific article