Approximation of conformal welding for finitely connected regions

We construct discrete conformal welding for finitely connected regions based on the circle packing approach. We show that the discrete conformal welding mappings induced by circle packings converge uniformly on compact subsets to their continuous counterparts and that the corresponding discrete conformal welding curves converge uniformly to quasicircles determined by quasisymmetric mappings. This gives a constructive proof of the existence and uniqueness theorem for conformal welding of finitely connected regions.