A mathematical study of the charge simulation method by use of peripheral conformal mappings
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 , ) 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 , 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 Riesz-Schauder theory to analyze it.