## Conformal moduli of symmetric circular quadrilaterals with cusps

Harri Hakula, Semen Nasyrov, and Matti Vuorinen

### Abstract

We investigate moduli of planar circular quadrilaterals that are symmetric with respect to both coordinate axes. First we develop an analytic approach that reduces this problem to ODEs and then devise a numerical method to find out the accessory parameters. This method uses the Schwarz equation to determine a conformal mapping of the unit disk onto a given circular quadrilateral. We also give an example of a circular quadrilateral for which the value of the conformal modulus can be found in analytic form. This example is used to validate the numeric calculations. We also apply another method, the so called hpFEM, for the numerical calculation of the moduli. These two different approaches provide results agreeing with high accuracy.

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### Key words

conformal capacity, conformal modulus, quadrilateral modulus, $hp$-FEM, numerical conformal mapping

### AMS subject classifications

65E05, 31A15, 30C85

### Links to the cited ETNA articles

 [30] Vol. 48 (2018), pp. 462-478 Harri Hakula, Antti Rasila, and Matti Vuorinen: Conformal modulus and planar domains with strong singularities and cusps

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