A Completely Iterative Method for the Infinite Domain Electrostatic Problem with Nonlinear Dielectric Media

This is the preprint of an article that will appear in the Journal of Computational Physics (DOI: 10.1016/j.jcp.2011.07.001).

A Completely Iterative Method for the Infinite Domain Electrostatic Problem with Nonlinear Dielectric Media

by Lun Yang and Kaushik Dayal, Carnegie Mellon University

Abstract

We present an iterative method for the solution of the exterior all-space electrostatic problem for nonlinear dielectric media. The electric potential is specified on interior boundaries and the electric field decays at infinity. Our approach uses a natural variational formulation based on the total energy of the nonlinear dielectric medium subject to boundary conditions. The problem is decomposed into an exterior calculation and an interior calculation with the boundary-specified electric potentials imposed as constraints between them. Together, these enable an iterative method that is based on the variational formulation. In contrast to direct solution of the electrostatic problems, we avoid the construction, storage and solution of dense and large linear systems. This provides important advantages for multiphysics problems that couple the linear electrostatic Poisson problem to nonlinear physics: the latter necessarily involves iterative approaches, and our approach replaces a large number of direct solves for the electrostatics with an iterative algorithm that can be coupled to the iterations of the nonlinear problem. We present examples applying the method to inhomogeneous, anisotropic nonlinear dielectrics. A key advantage of our variational formulation is that we require only the free-space, isotropic, homogeneous Greens function for all these settings.