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Die Kollokationsmethode angewandt auf die Symmsche Integralgleichung. (German) Zbl 0579.65142
Eidgenössische Technische Hochschule Zürich. 159 S. (1983).
Summary: The conformal mapping of a simply connected bounded Jordan region onto the unit disk may be described by a boundary correspondence function. For the derivative of such a function, a relationship is given by Symm’s integral equation. This work deals with numerical methods to approximate this boundary correspondence function. First we show that for a region with continuously differentiable boundary curve Symm’s integral operator defines an isomorphism from \(L_ 2\) into the space of absolutely continuous functions. To discretize Symm’s integral equation, we discuss three methods: the collocation method, the Galerkin method and the least squares approximation method; in all cases the boundary correspondence function is approximated by splines functions. We obtain optimal rates of convergence - the same rates as for the best approximation. Moreover, we have the superapproximation property - optimal convergence in weaker norms than the \(L_ 2\)-norm. Finally, the collocation method is applied to some test regions (ellipse, reflected ellipse, capsule). We notice that the asymptotic error estimates are in good agreement with the numerical results.

65R20 Numerical methods for integral equations
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
30C30 Schwarz-Christoffel-type mappings
45H05 Integral equations with miscellaneous special kernels