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.
| Original language | English |
|---|---|
| Pages (from-to) | 460-482 |
| Number of pages | 23 |
| Journal | Electronic Transactions on Numerical Analysis |
| Volume | 54 |
| DOIs | |
| Publication status | Published - 2021 |
| MoE publication type | A1 Journal article-refereed |
Funding
∗Received February 6, 2021. Accepted May 16, 2021. Published online on May 30, 2021. Recommended by Tom De Lillo. The work of the second author is supported by the development program of the Volga Region Mathematical Center (agreement no. 075-02-2021-1393). †Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland ([email protected]) ‡Kazan Federal University, Kazan, Russia ([email protected]). §Department of Mathematics and Statistics, FI-20014 University of Turku, Finland ([email protected]).
Keywords
- Conformal capacity
- Conformal modulus
- Hp-FEM
- Numerical conformal mapping
- Quadrilateral modulus