Volume 48, pp. 462-478, 2018.

Conformal modulus and planar domains with strong singularities and cusps

Harri Hakula, Antti Rasila, and Matti Vuorinen


We study the problem of computing the conformal modulus of rings and quadrilaterals with strong singularities and cusps at their boundary. We reduce this problem to the numerical solution of the associated Dirichlet and Dirichlet-Neumann-type boundary values problems for the Laplace equation. Several experimental results, with error estimates, are reported. In particular, we consider domains with dendrite-like boundaries where an analytic formula for the conformal modulus can be derived. The boundary value problems are solved using an $hp$-finite element method.

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Key words

conformal capacity, conformal modulus, quadrilateral modulus, $hp$-FEM, numerical conformal mapping

AMS subject classifications

65E05, 31A15, 30C85

Links to the cited ETNA articles

[15]Vol. 40 (2013), pp. 436-451 Harri Hakula, Antti Rasila, and Matti Vuorinen: Computation of exterior moduli of quadrilaterals

ETNA articles which cite this article

Vol. 54 (2021), pp. 460-482 Harri Hakula, Semen Nasyrov, and Matti Vuorinen: Conformal moduli of symmetric circular quadrilaterals with cusps

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