Series Expansions of the Layer Potential Operators Using the Faber Polynomials and Their Applications to the Transmission Problem
Younghoon Jung, Mikyoung Lim
Abstract
We consider the conductivity transmission problem in two dimensions with a simply connected inclusion of arbitrary shape. It is well known that the solvability of the transmission problem can be established via the boundary integral formulation in which the Neumann--Poincaré (NP) operator is involved. In this paper, we derive series expansions of the layer potential operators based on geometric function theory and exhibit a novel approach to the transmission problem. We first construct a collection of harmonic basis functions by using the Faber polynomials associated with the simply connected inclusion. We then derive explicit series expansions of the layer potential operators with respect to the constructed basis functions. In particular, the NP operator becomes a doubly infinite, self-adjoint matrix operator, whose entry is explicitly given by the Grunsky coefficients corresponding to the inclusion. By applying the finite section method to this matrix formulation, we obtain an approximation scheme for the spectrum of the NP operator and the solution to the transmission problem. This scheme allows us to numerically compute, in a simple way, the spectrum of the NP operator for a smooth domain when the exterior conformal mapping is known. We additionally derive an explicit integral expression for the coefficients of the exterior conformal mapping so that for an inclusion of arbitrary shape, one can numerically compute the exterior conformal mapping by solving only one boundary integral equation and apply the proposed approach. We provide numerical examples to demonstrate the effectiveness of the proposed method.