Fast capacity computation for maze-like configurations

TL;DR

This paper develops a high-order finite element method to compute the conformal capacity of maze-like structures and compares these results with estimates based on quasihyperbolic length and perimeter, demonstrating efficiency and accuracy.

Contribution

It introduces a novel high-order $hp$- finite element approach for capacity computation in maze-like domains and evaluates the effectiveness of quasihyperbolic estimates.

Findings

Quasihyperbolic estimates show good asymptotic behavior.

The finite element method provides accurate capacity calculations.

Quasihyperbolic estimates are computationally efficient.

Abstract

We study the conformal capacity ${\rm cap}(\Omega,K)$ where $\Omega$ is a bounded domain of $\mathbb{R}^2$ and $K$ is a compact connected set in $\Omega$. Because the exact numerical value of the capacity is known only in a handful of special cases, it is important to find estimates for the capacity in terms of domain functionals, simpler than the capacity itself. Here, we study condensers of maze-like structure and compute their capacity by means of a high-order $hp$- finite element method. We compare these numerical results to the estimates given by the quasihyperbolic length and perimeter of the compact set. In particular, we consider the behaviour of these value pairs, numerical results and estimates, when the structure parameters vary and the walls of the maze approach the compact set. Over the configurations covered in the numerical experiments, the quasihyperbolic estimates are shown to have the desired asymptotic properties and superior computational efficiencies once the case-specific analysis is completed.