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

On moduli of rings and quadrilaterals: algorithms and experiments