On the polarizability and capacitance of the cube

TL;DR

This paper introduces an efficient integral equation solver for calculating the electrostatic polarizability and capacitance of cuboidal domains, providing highly accurate results and a rigorous mathematical framework for Lipschitz domains.

Contribution

It develops a new integral equation method for electrostatics on cuboidal domains and establishes a rigorous mathematical framework for polarizability of Lipschitz domains.

Findings

Accurate polarizability calculations for dielectric cubes at various permittivities.

Capacitance of the unit cube determined with unprecedented precision.

Mathematical insights into the behavior of polarizability measures.

Abstract

An efficient integral equation based solver is constructed for the electrostatic problem on domains with cuboidal inclusions. It can be used to compute the polarizability of a dielectric cube in a dielectric background medium at virtually every permittivity ratio for which it exists. For example, polarizabilities accurate to between five and ten digits are obtained (as complex limits) for negative permittivity ratios in minutes on a standard workstation. In passing, the capacitance of the unit cube is determined with unprecedented accuracy. With full rigor, we develop a natural mathematical framework suited for the study of the polarizability of Lipschitz domains. Several aspects of polarizabilities and their representing measures are clarified, including limiting behavior both when approaching the support of the measure and when deforming smooth domains into a non-smooth domain. The success of the mathematical theory is achieved through symmetrization arguments for layer potentials.