## On moduli of rings and quadrilaterals: algorithms and experiments.(English)Zbl 1368.65036

Summary: Moduli of rings and quadrilaterals are frequently applied in geometric function theory; see, e.g., the handbook by R. Kühnau [Handbook complex analysis: geometric function theory. Vols. 1 and 2. Amsterdam: North-Holland (2005; Zbl 1057.30001, Zbl 1056.30002)]. 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.

### MSC:

 65E05 General theory of numerical methods in complex analysis (potential theory, etc.) 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions 30C85 Capacity and harmonic measure in the complex plane

### Citations:

Zbl 1057.30001; Zbl 1056.30002

### Software:

Schwarz-Christoffel; SC Toolbox; CONFPACK; DLMF
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