A Simple Numerical Method for Complex Geometrical Optics Solutions to the Conductivity Equation

Abstract

This paper concerns numerical methods for computing complex geometrical optics (CGO) solutions to the conductivity equation $\nabla\cdot\sigma\nabla u(\cdot,k)=0$ in $\mathbb{R}^2$ for piecewise smooth conductivities $\sigma$, where k is a complex parameter. The key is to solve an $\mathbb R$-linear singular integral equation defined in the unit disk. Recently, Astala et al. [Appl. Comput. Harmon. Anal., 29 (2010), pp. 2–17] proposed a complicated method for numerical computation of CGO solutions by solving a periodic version of the $\mathbb{R}$-linear integral equation in a rectangle containing the unit disk. In this paper, based on the fast algorithms in [P. Daripa and D. Mashat, Numer. Algorithms, 18 (1998), pp. 133–157] for singular integral transforms, we propose a simpler numerical method which solves the $\mathbb{R}$-linear integral equation in the unit disk directly. For the resulting $\mathbb{R}$-linear operator equation, a minimal residual iterative method is proposed. Numerical examples illustrate the accuracy and efficiency of the new method.