Conformal moduli of symmetric circular quadrilaterals with cusps

Harri Hakula; Semen Nasyrov; Matti Vuorinen

doi:10.1553/ETNA_VOL54S460

Conformal moduli of symmetric circular quadrilaterals with cusps

Harri Hakula
, Semen Nasyrov
, Matti Vuorinen

Tutkimustuotos: Lehtiartikkeli › Article › Scientific › vertaisarvioitu

4 Sitaatiot (Scopus)

266 Lataukset (Pure)

Abstrakti

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.

Alkuperäiskieli	Englanti
Sivut	460-482
Sivumäärä	23
Julkaisu	Electronic Transactions on Numerical Analysis
Vuosikerta	54
DOI - pysyväislinkit	https://doi.org/10.1553/ETNA_VOL54S460
Tila	Julkaistu - 2021
OKM-julkaisutyyppi	A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

Rahoitus

∗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]).

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title = "Conformal moduli of symmetric circular quadrilaterals with cusps",

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.",

keywords = "Conformal capacity, Conformal modulus, Hp-FEM, Numerical conformal mapping, Quadrilateral modulus",

author = "Harri Hakula and Semen Nasyrov and Matti Vuorinen",

note = "Funding Information: ∗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]). Publisher Copyright: {\textcopyright} 2021, Kent State University.",

year = "2021",

doi = "10.1553/ETNA\_VOL54S460",

language = "English",

volume = "54",

pages = "460--482",

journal = "Electronic Transactions on Numerical Analysis",

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T1 - Conformal moduli of symmetric circular quadrilaterals with cusps

AU - Hakula, Harri

AU - Nasyrov, Semen

AU - Vuorinen, Matti

N1 - Funding Information: ∗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]). Publisher Copyright: © 2021, Kent State University.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Conformal capacity

KW - Conformal modulus

KW - Hp-FEM

KW - Numerical conformal mapping

KW - Quadrilateral modulus

UR - https://www.scopus.com/pages/publications/85109911200

U2 - 10.1553/ETNA_VOL54S460

DO - 10.1553/ETNA_VOL54S460

M3 - Article

AN - SCOPUS:85109911200

SN - 1068-9613

VL - 54

SP - 460

EP - 482

JO - Electronic Transactions on Numerical Analysis

JF - Electronic Transactions on Numerical Analysis

ER -

Conformal moduli of symmetric circular quadrilaterals with cusps

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