A mathematical study of the charge simulation method by use of peripheral conformal mappings
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Abstract
In this paper, we discuss a convergence theorem on charge simulation method (CSM) which is a numerical algorithm for solving boundary value problem of the Laplace equation. CSM requires us to select two kinds of points. They are collocation points (we denote them by [numerical formula]) and charge points (we denote them by [numerical formula]). Though choosing these points appropriately is a fundamental problem, no good rules were known until quite recently. For a two dimensional Jordan region Ω with an analytic boundary Γ, the author dealt with a rule (Katsurada [10], [10]) in which we use mapping function of the region to determine charge points and collocation points. But it was not satisfactory, since finding a concrete mapping function is not so easy. In our recent paper Katsurada and Okamoto [8], we have proposed a new rule to determine these points, introducing a kind of peripheral conformal mapping Ψ with (i) Ψ maps the unit circle to Γ: [numerical formula], and (ii) Ψ is conformal in a neighborhood of the unit circle: [numerical formula]. Using such Ψ, we select collocation points and charge points by [numerical formula], where [numerical formula]. The purpose of this paper is to state a convergence theorem concerning the new rule, and prove it. We regard CSM as a discretization of solution by a generalized integral equation on the boundary Γ, and Γ is a perturbation of a unit circle, and we use the RieszSchauder theory to analyze it.
Journal

 Memoirs of the Institute of Sciences and Technology, Meiji University

Memoirs of the Institute of Sciences and Technology, Meiji University 37, 195211, 1998
Meiji University