On the numerical approximation of unstable minimal surfaces with polygonal boundaries

Summary. This work is concerned with the approximation and the numerical computation of polygonal minimal surfaces in ${\Bbb R}^q (q \ge 2)$ . Polygonal minimal surfaces correspond to the critical points of Shiffman's function $\Theta$ . Since this function is analytic, polygonal minimal surfaces can be characterized by means of the second derivative of $\Theta$. We present a finite element approximation of quasiminimal surfaces together with an error estimate. In this way we obtain discrete approximations $\Theta_h$ of $\Theta$ and $f_h$ of $\nabla \Theta$ . 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 $f_h$.

[1]  Richard Courant.  Dirichlet’s Principle and the Boundary Value Problem of Potential Theory , 1950 .

[2]  On the classification of unstable minimal surfaces with polygonal boundaries , 1955 .

[3]  Erhard Heinz.  Über eine Verallgemeinerung des Plateauschen Problems , 1979 .

[4]  Eine bemerkung zu den Marx-Shiffmanschen Minimalvektoren bei polygonen , 1981 .

[5]  Erhard Heinz.  Minimalflächen mit polygonalem Rand , 1983 .

[6]  Zum Marx-Shiffmanschen Variationsproblem. , 1983 .

[7]  Ein Eindeutigkeitssatz für Minimalflächen im Rp mit polygonalem Rand. , 1985 .

[8]  Friedrich Sauvigny.  On the Morse index of minimal surfaces in ℝp with polygonal boundaries , 1985 .

[9]  Takuya Tsuchiya,et al.  On two methods for approximating minimal surfaces in parametric form , 1986 .

[10]  Takuya Tsuchiya,et al.  Discrete solution of the Plateau problem and its convergence , 1987 .

[11]  Johannes C. C. Nitsche,et al.  Lectures on minimal surfaces: vol. 1 , 1989 .

[12]  G. Dziuk,et al.  An algorithm for evolutionary surfaces , 1990 .

[13]  Takuya Tsuchiya,et al.  A note on discrete solutions of the Plateau problem , 1990 .

[14]  J. Hutchinson.  Computing conformal maps and minimal surfaces , 1991 .

[15]  Robert Osserman,et al.  Lectures on Minimal Surfaces. , 1991 .

[16]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[17]  Michael Hinze.  On the Numerical Treatment of Quasiminimal Surfaces , 1993, IMPACT Comput. Sci. Eng..

[18]  J. Rappaz,et al.  Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems , 1994 .

[19]  Gerhard Dziuk,et al.  On the approximation of unstable parametric minimal surfaces , 1996 .