  We consider Riemann mappings from bounded Lipschitz domains in the plane to a
triangle. We show that in this case the Riemann mapping has a linear
variational principle: it is the minimizer of the Dirichlet energy over an
appropriate affine space. By discretizing the variational principle in a
natural way we obtain discrete conformal maps which can be computed by solving
a sparse linear system. We show that these discrete conformal maps converge to
the Riemann mapping in $H^1$, even for non-Delaunay triangulations.
Additionally, for Delaunay triangulations the discrete conformal maps converge
uniformly and are known to be bijective. As a consequence we show that the
Riemann mapping between two bounded Lipschitz domains can be uniformly
approximated by composing the Riemann mappings between each Lipschitz domain
and the triangle.
