On the Numerical Approximation of Unstable Minimal Surfaces with Polygonal Boundaries

This work is concerned with the approximation and the numerical computation of polygonal minimal surfaces in IR q (q 2). Polygonal minimal surfaces correspond to the critical points of Shiiman's function. Since this function is analytic, polygonal minimal surfaces can be characterized by means of the second derivative of. We present a nite element approximation of quasiminimal surfaces together with an error estimate. In this way we obtain discrete approximations h of and fh of r. In particular we prove that the discrete functions converge uniformly on certain compact subsets. This will be the main tool for proving existence and convergence of discrete minimal surfaces in neighbourhoods of non-degenerate minimal surfaces. In the numerical part of this paper we compute numerical approximations of polygonal minimal surfaces by use of Newton's method applied to fh.