Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 139

On the Numerical Approximation of Unstable Minimal Surfaces with Polygonal Boundaries

M. Hinze

This work is concerned with the approximation and the numerical computation of polygonal minimal surfaces in the euklidean q-space. Polygonal minimal surfaces correspond to the critical points of Shiffman`s function. Since this function is analytic, polygonal minimal surfaces can be characterized by means of the second derivative of this function. We present a finite element approximation of quasiminimal surfaces together with an error estimate. In this way we obtain discrete approximations of Shiffman`s function and the gradient of Shiffman`s function. 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.

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