L²-estimate for the discrete Plateau Problem

Abstract

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.