On moduli of rings and quadrilaterals: Algorithms and experiments

Harri Hakula; Antti Rasila; Matti Vuorinen

doi:10.1137/090763603

On moduli of rings and quadrilaterals: Algorithms and experiments

Harri Hakula^*, Antti Rasila, Matti Vuorinen

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

31 Scopus citations

Abstract

Moduli of rings and quadrilaterals are frequently applied in geometric function theory; see, e.g., the handbook by Kühnau [Handbook of Complex Analysis: Geometric Function Theory, Vols. 1 and 2, North-Holland, Amsterdam, 2005]. 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 nonpolygonal boundary and report results with concrete error bounds.

Original language	English
Pages (from-to)	279-302
Number of pages	24
Journal	SIAM Journal of Scientific Computing
Volume	33
Issue number	1
DOIs	https://doi.org/10.1137/090763603
State	Published - 2011
Externally published	Yes

Keywords

Conformal capacity
Conformal modulus
Hp-FEM
Numerical conformal mapping
Quadrilateral modulus

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10.1137/090763603

Cite this

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AU - Hakula, Harri

AU - Rasila, Antti

AU - Vuorinen, Matti

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N2 - Moduli of rings and quadrilaterals are frequently applied in geometric function theory; see, e.g., the handbook by Kühnau [Handbook of Complex Analysis: Geometric Function Theory, Vols. 1 and 2, North-Holland, Amsterdam, 2005]. 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 nonpolygonal boundary and report results with concrete error bounds.

AB - Moduli of rings and quadrilaterals are frequently applied in geometric function theory; see, e.g., the handbook by Kühnau [Handbook of Complex Analysis: Geometric Function Theory, Vols. 1 and 2, North-Holland, Amsterdam, 2005]. 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 nonpolygonal boundary and report results with concrete error bounds.

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KW - Conformal modulus

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KW - Numerical conformal mapping

KW - Quadrilateral modulus

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On moduli of rings and quadrilaterals: Algorithms and experiments

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Keywords

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