Discrete harmonic maps and convergence to conformal maps, I: Combinatorial harmonic coordinates

In this paper, we provide new discrete uniformization theorems for bounded, m-connected planar domains. To this end, we consider a planar, bounded, m-connected domain Ω, and let δΩ be its boundary. Let T denote a triangulation of Ω ∪ δΩ. We construct a new decomposition of Ω ∪ δΩ into a finite union of quadrilaterals with disjoint interiors. The construction is based on utilizing a pair of harmonic functions on T(0) and properties of their level curves. In the sequel [26], it will be proved that a particular discrete scheme based on these theorems converges to a conformal map, thus providing an affirmative answer to a conjecture raised by Stephenson [41, Section 11].