Corner effects on the perturbation of an electric potential

TL;DR

This paper introduces new geometric factors related to corner effects on electric potential perturbations caused by inclusions, linking them to polarization tensors and providing criteria for corner existence with numerical validation.

Contribution

It defines geometric factors based on conformal mappings that characterize corner effects and relate to polarization tensors, offering a new analytical approach for boundary analysis.

Findings

Geometric factors are Fourier coefficients of a generalized external angle.

Criteria for corner existence are derived from geometric factors.

Numerical methods validate the theoretical results with high precision.

Abstract

We consider the perturbation of an electric potential due to an insulating inclusion with corners. This perturbation is known to admit a multipole expansion whose coefficients are linear combinations of generalized polarization tensors. We define new geometric factors of a simple planar domain in terms of a conformal mapping associated with the domain. The geometric factors share properties of the generalized polarization tensors and are the Fourier series coefficients of a kind of generalized external angle of the inclusion boundary. Since the generalized external angle contains the Dirac delta singularity at corner points, we can determine the criterion for the existence of corner points on the inclusion boundary in terms of the geometric factors. We illustrate and validate our results with numerical examples computed to a high degree of precision using integral equation techniques, Nystr\"om discretization, and recursively compressed inverse preconditioning.