The discrete plateau problem: Convergence results

Abstract

We solve the problem of finding and justifying an optimal fully discrete finite element procedure for approximating minimal, including unsta- ble, surfaces. In this paper we introduce the general framework and some preliminary estimates, develop the algorithm, and give the numerical results. In a subsequent paper we prove the convergence estimate. The algorithmic procedure is to find stationary points for the Dirichlet en- ergy within the class of discrete harmonic maps from the discrete unit disc such that the boundary nodes are constrained to lie on a prescribed boundary curve. An integral normalisation condition is imposed, corresponding to the usual three point condition. Optimal convergence results are demonstrated numerically and theoretically for nondegenerate minimal surfaces, and the ne- cessity for nondegeneracy is shown numerically.