We propose the complex dipole simulation method (CDSM) which approximates a holomorphic function by linear combination of 1/(z − ζ ) with the use of its boundary values. In this paper, we treat a function f which is holomorphic in Ω and continuous on Ω in the case where Ω is a disk or the exterior domain of a disk.Then we establish the following fact: if f is holomorphic in some neighborhood of Ω, the error of an approximate function f (N) decays exponentially with respect to N , where N is the number of the charge points.
In this study, the solutions to the one‐phase interior or the classical Hele‐Shaw problem are discretized in space by employing the method of fundamental solutions combined with the uniform distribution method; furthermore, a system of ordinary differential equations is obtained, which is solved using the usual fourth‐order Runge‐Kutta method. The one‐phase interior Hele‐Shaw problem has curve‐shortening (CS), area‐preserving (AP), and barycenter‐fixed (BF) properties. Under our numerical scheme, a discrete version of CS and BF properties holds in an asymptotic sense and AP property in an exact sense, whereas in general, simple boundary element method does not satisfy these properties. Moreover, the one‐phase exterior Hele‐Shaw problem and one‐phase interior Hele‐Shaw problem containing sink/source points can be treated. Therefore, in each problem, a nontrivial exact solution is constructed and an experimental order of convergence is shown.
The aim of this paper is to develop the method of fundamental solutions using weighted average condition and dummy points. We accomplish mathematical analysis, a unique existence of an approximate solution and an exponential decay of the approximation error, for a potential problem in disk, and show some numerical experiments, which exemplify our error estimate.
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