Die Kollokationsmethode angewandt auf die Symmsche Integralgleichung. (German) Zbl 0579.65142

Hoidn, Hans-Peter

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.

MSC:

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

Keywords:

conformal mapping; Symm’s integral equation; collocation method; Galerkin method; least squares; optimal rates of convergence; best approximation; asymptotic error estimates; numerical results