# Computation of conformal invariants

## TL;DR

This paper presents a boundary integral equation-based algorithm for numerically computing conformal invariants, specifically conformal capacity, of planar domains with various boundary smoothness, demonstrating accuracy and versatility.

## Contribution

It introduces a flexible boundary integral method for computing conformal capacity across diverse geometries, with implementation details and performance comparisons.

## Key findings

- Algorithm effectively computes conformal capacity for smooth and piecewise smooth domains.
- Method shows high accuracy compared to existing results.
- Versatile approach applicable to various geometrical configurations.

## Abstract

We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains are assumed to have smooth or piecewise smooth boundaries. The method we use is based on the boundary integral equation method developed and implemented in [30]. A characteristic feature of this method is that, with small changes in the code, a wide spectrum of problems can be treated and we include code snippets within the text to indicate implementation details. We compare the performance and accuracy to previous results in the cases when numerical data is available and also in the case of several model problems where exact results are available.