Fast Computation of Electrostatic Potentials for Piecewise Constant Conductivities

We present a novel numerical method for solving the elliptic partial differential equation problem for the electrostatic potential with piecewise constant conductivity. We employ an integral equation approach for which we derive a system of well-conditioned integral equations by representing the solution as a sum of single layer potentials. The kernel of the resulting integral operator is smooth provided that the layers are well-separated. The fast multiple method is used to accelerate the generalized minimal residual method solution of the integral equations. For efficiency, we adapt the grid of the Nystr\"{o}m method based on the spectral resolution of the layer charge density. Additionally, we present a method for evaluating the solution that is efficient and accurate throughout the domain, circumventing the close-evaluation problem. To support the design choices of the numerical method, we derive regularity estimates with bounds explicitly in terms of the conductivities and the geometries of the boundaries between their regions. The resulting method is fast and accurate for solving for the electrostatic potential in media with piecewise constant conductivities.