Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
ACM Transactions on Mathematical Software (TOMS)
A Charge Simulation Method for Numerical Conformal Mapping onto Circular and Radial Slit Domains
SIAM Journal on Scientific Computing
An Interpolating Polynomial Method for Numerical Conformal Mapping
SIAM Journal on Scientific Computing
2D-Shape Analysis Using Conformal Mapping
International Journal of Computer Vision
Convergence of a Variant of the Zipper Algorithm for Conformal Mapping
SIAM Journal on Numerical Analysis
Revisiting the Crowding Phenomenon in Schwarz-Christoffel Mapping
SIAM Journal on Scientific Computing
NIST Handbook of Mathematical Functions
NIST Handbook of Mathematical Functions
Conjugate function method for numerical conformal mappings
Journal of Computational and Applied Mathematics
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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.