A simple numerical method for complex geometrical optics solutions to the conductivity equation

This paper concerns numerical methods for computing complex geometrical optics (CGO) solutions to the conductivity equation ∇· σ∇u(·, κ) = 0 in ℝ² for piecewise smooth conductivities σ, where κ is a complex parameter. The key is to solve an 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 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 ℝ-linear integral equation in the unit disk directly. For the resulting R-linear operator equation, a minimal residual iterative method is proposed. Numerical examples illustrate the accuracy and efficiency of the new method.