Combinatorial Harmonic Coordinates

Consider a planar, bounded, $m$-connected domain $\Omega$ and let $\partial\Omega$ be its boundary. Let $\mathcal{T}$ denote a cellular decomposition of $\Omega\cup\partial\Omega$ where each $2$-cell is either a triangle or a quadrilateral. We construct a \emph{new} decomposition of $\Omega\cup\partial\Omega$ into ${\mathcal R}$, a finite, disjoint union of quadrilaterals. The construction is based on utilizing a {\it pair} of functions on ${\mathcal T}^{(0)}$ and properties of their level curves.

The first function is obtained as the solution of a Dirichlet boundary value problem defined on ${\mathcal T}^{(0)}$. The second function is obtained by an integration scheme along the level curves of the first, and is called the {\it conjugate} function.
It turns out that the pair of functions can be made {\it orthogonal} and {\it harmonic} in a combinatorial sense. The applications include new discrete
uniformization theorems for bounded, $m$-connected planar domains.