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.

翻译：共轭函数方法是用于单连通及多连通区域数值共形映射计算的算法。本文将该方法推广至黎曼曲面间的共形映射。核心挑战在于建立曲面上拉普拉斯-贝尔特拉米方程与四边形共形模量计算之间的关联。我们考虑了单连通、双连通及多连通区域的映射，数值计算基于$hp$自适应有限元方法。该方法的关键优势在于能够对包含强奇异性和尖点等复杂边界几何的区域实现高精度曲面映射。通过包含误差估计在内的大量数值实验验证了所提方法的有效性。