L2-estimate for the discrete Plateau Problem

In this paper we prove the L2 convergence rates for a fully discrete finite element procedure for approximating minimal, possibly unstable, surfaces. Originally this problem was studied by G. Dziuk and J. Hutchinson. First they provided convergence rates in the H1 and L2 norms for the boundary integral method. Subsequently they obtained the H1 convergence estimates using a fully discrete finite element method. We use the latter framework for our investigation.