Before revision:  We show that these discrete conformal maps converge to the Riemann mapping uniformly and in H^1  . Additionally,  the discrete conformal maps  are known to be bijective , and we show that the Riemann mapping between two bounded Lipschitz domains can be approximated arbitrarily well by composing the Riemann mapping between each Lipschitz domain and the triangle.
After revision:  We show that these discrete conformal maps converge to the Riemann mapping uniformly and in H^1 , even for non-Delaunay triangulations . Additionally, the discrete conformal maps are known to be bijective , and we show that the Riemann mapping between two bounded Lipschitz domains can be approximated arbitrarily well by composing the Riemann mapping between each Lipschitz domain and the triangle.