Efficient Numerical Conformal Mappings on Multiply Connected Riemann Surfaces

Abstract

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains on surfaces. In this paper the conjugate function method, earlier used for simply connected domains, is generalized and refined to achieve the same level of accuracy on multiply connected planar domains and Riemann surfaces. The main challenge is the accurate and efficient construction of boundary values for the conjugate problem on multiply connected domains. The method relies on high-order finite element methods which allow for highly accurate computations of mappings on surfaces, including domains of complex boundary geometry containing strong singularities and cusps. We also derive the reciprocal error estimate for the multiply connected case. The efficacy of the proposed method is illustrated via an extensive set of numerical experiments with error estimates.