The discrete Douglas problem: Theory and numerics

We solve the problem of finding and justifying an optimal fully discrete finite element procedure for approximating annulus-like, possibly unstable, minimal surfaces. In this paper we introduce the general framework, some preliminary estimates, develop the ideas used for the algorithm, and give the numerical results. Similarities and differences with respect to the fully discrete finite element procedure given by G. Dziuk and J. Hutchinson in the case of the classical Plateau problem are also addressed. In a subsequent paper we prove convergence estimates.