On moduli of rings and quadrilaterals: algorithms and experiments

TL;DR

This paper introduces a new $hp$-FEM algorithm for accurately computing the moduli of rings and quadrilaterals, including non-polygonal cases, and compares its performance with existing methods.

Contribution

The paper presents a novel $hp$-FEM algorithm for moduli computation that extends to non-polygonal boundaries and provides concrete error bounds.

Findings

$hp$-FEM algorithm achieves high accuracy in moduli computation.

Algorithm outperforms existing methods like Schwarz-Christoffel Toolbox.

Effective for non-polygonal boundary domains.

Abstract

Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by K\"uhnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new $hp$-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the $hp$-FEM algorithm applies to the case of non-polygonal boundary and report results with concrete error bounds.