Conformal capacity and polycircular domains

TL;DR

This paper develops numerical methods to compute conformal capacities of multiply connected polycircular domains, combining boundary integral equations with fast multipole and $hp$-FEM methods, validated by error estimates and convergence analysis.

Contribution

It introduces a combined computational approach using boundary integral equations and $hp$-FEM for accurate conformal capacity calculations of polycircular domains.

Findings

High-accuracy agreement between the two methods

Error estimates provided for computed capacities

Convergence analysis under discretization refinement

Abstract

We study numerical conformal mapping of multiply connected planar domains with boundaries consisting of unions of finitely many circular arcs, so called polycircular domains. We compute the conformal capacities of condensers defined by polycircular domains. Experimental error estimates are provided for the computed capacity and, when possible, the rate of convergence under refinement of discretisation is analysed. The main ingredients of the computation are two computational methods, on one hand the boundary integral equation method combined with the fast multipole method and on the other hand the $hp$-FEM method. The results obtained with these two methods agree with high accuracy.