Hybrid Basis Scheme for computing Electrostatic fields exterior to close-to-touching discs

TL;DR

This paper introduces a new hybrid basis numerical scheme for efficiently computing electrostatic fields around close-to-touching discs, with proven faster convergence and easy adaptability to multiple cylinders.

Contribution

A novel hybrid basis method combining Fourier-Laurent and conformal mapping expansions for electrostatic problems involving close-to-touching discs.

Findings

Faster decay rate of basis coefficients with the hybrid method.

Rigorous proof of the hybrid basis's effectiveness.

Easy implementation and adaptability to multiple cylinders.

Abstract

This paper presents a simple and effective new numerical scheme for the computation of electrostatic fields exterior to a collection of close-to-touching discs. The method is presented in detail for the two-cylinder case. The key idea is to represent the required complex potential using a hybrid set of basis functions comprising the usual Fourier-Laurent expansion about each circle centre comple- mented by a subsidiary expansion in a variable associated with conformal mapping of the physical domain to a concentric annulus domain. We also rigorously prove that there is a representation of the solution in the hybrid basis with faster decay rate of coefficients than is obtained by using a non-hybrid basis, thereby providing a rationalization for the success of the method. The numerical scheme is easy to implement and adaptable to the case of multiple close-to-touching cylinders.