Laplace--Beltrami Equations and Numerical Conformal Mappings on Surfaces

TL;DR

This paper extends the conjugate function method to compute conformal mappings on Riemannian surfaces using an hp-adaptive finite element method, enabling highly accurate mappings even with complex geometries.

Contribution

It introduces a novel approach connecting Laplace--Beltrami equations with conformal mapping computations on surfaces, including complex boundary features.

Findings

Accurate numerical conformal mappings on complex surfaces.

Effective handling of singularities and cusps.

Validated through extensive numerical experiments.

Abstract

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains. In this paper, the conjugate function method is extended to cover conformal mappings between Riemannian surfaces. The main challenge addressed here is the connection between Laplace--Beltrami equations on surfaces and the computation of the conformal modulus of a quadrilateral. We consider mappings of simply, doubly, and multiply connected domains. The numerical computation is based on an $hp$-adaptive finite element method. The key advantage of our approach is that it allows highly accurate computations of mappings on surfaces, including domains of complex boundary geometry involving strong singularities and cusps. The efficacy of the proposed method is illustrated via an extensive set of numerical experiments including error estimates.