Solution of the Björling problem by discrete approximation

The Bj\"orling problem amounts to the construction of a minimal surface from a real-analytic curve with a given real-analytic normal vector field. We approximate that solution locally by discrete minimal surfaces as special discrete isothermic surfaces (as defined by Bobenko and Pinkall in 1996). The main step in our construction is the approximation of the sought surface's Weierstrass data by discrete conformal maps. We prove that the approximation error is of the order of the square of the mesh size.