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The discrete Plateau Problem:
Algorithm and numerics


Authors: Gerhard Dziuk and John E. Hutchinson
Journal: Math. Comp. 68 (1999), 1-23
MSC (1991): Primary 65N30; Secondary 49Q05, 53A10
DOI: https://doi.org/10.1090/S0025-5718-99-01025-X
MathSciNet review: 1613695
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Abstract: We solve the problem of finding and justifying an optimal fully discrete finite element procedure for approximating minimal, including unstable, 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 energy 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 necessity for nondegeneracy is shown numerically.


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  • [A1] H.W. Alt, Verzweigungspunkte von H-Flächen I, Math. Zeit. 127 (1972), 333-362. MR 47:965b
  • [A2] H.W. Alt, Verzweigungspunkte von H-Flächen II, Math. Ann. 201 (1973), 33-55. MR 48:9529
  • [Br] K.A. Brakke, The Surface Evolver, Exp. Math. 1 (1992), 141-165. MR 93k:53006
  • [BT] R. Böhme & T. Tromba, The Index Theorem for Classical Minimal Surfaces, Ann. Math. 113 (1981), 447-499. MR 83a:58031
  • [Ci] P.G. Ciarlet, The Finite Element Methods for Elliptic Problems, North Holland 1978. MR 58:25001
  • [Co] P. Concus, Numerical Solution of the Minimal Surface Equation, Math. Comp. 21 (1967), 340-350. MR 37:4968
  • [Cou] R. Courant, Dirichlet's Principle, Conformal Mapping and Minimal Surfaces, Interscience 1950. MR 12:90a
  • [Dou] J. Douglas, Solution of the Problem of Plateau, Trans. Am. Math. Soc. 33 (1931), 263-321.
  • [Dou2] J. Douglas, A Method of Numerical Solution of the Plateau Problem, Ann. Math. (2) 29 (1928), 180-188.
  • [Dz] G. Dziuk, An Algorithm for Evolutionary Surfaces, Num. Math. 58 (1991), 603-611. MR 91k:65042
  • [DH0] G. Dziuk, J.E. Hutchinson, On the approximation of unstable parametric minimal surfaces, preprint No. 340 (1994) SFB 256, Bonn, or CMA Math. Res. Rep. 9 (1994), Australian National University.
  • [DH1] G. Dziuk, J.E. Hutchinson, On the approximation of unstable parametric minimal surfaces, Calc. Var. 4 (1996), 27-58. MR 96m:49073
  • [DH2] G. Dziuk, J.E. Hutchinson, $L^2$ Estimates for Approximations to Minimal Surfaces, Proceedings of the International Conference: Curvature Flows and Related Topics, 67-92, A. Damlamian, J. Spruck & A. Visintin eds., Gakkotosho, Tokyo, 1995. MR 97g:58042
  • [DH3] G. Dziuk, J.E. Hutchinson, A Finite Element Method for the Computation of Parametric Minimal Surfaces, Tatra Mount. Math. J. 4 (1994), 49-62. MR 95g:65151
  • [DH4] G. Dziuk, J.E. Hutchinson, The Discrete Plateau Problem: Convergence Results, to appear.
  • [DHKW] U. Dierkes, S. Hildebrandt, A. Küster & O. Wohlrab, Minimal Surfaces I & II, Grundlehren der Mathematischen Wissenschaften 295-6, Springer-Verlag 1992. MR 94c:49001a; MR 94c:49001b
  • [G] R. Gulliver, Regularity of Minimizing Surfaces of Prescribed Mean Curvature, Ann. Math. 97 (1973), 275-305. MR 47:5736
  • [He] E. Heinz, Über das Randverhalten quasilinearer elliptischer Systeme mit isothermen Parametern, Math. Zeit. 113 (1970), 99-105. MR 41:7288
  • [Hi1] M. Hinze, On the Numerical Treatment of Quasiminimal Surfaces, Preprint 315 TU Berlin 1992. MR 95b:49063
  • [Hi2] M. Hinze, On a Simple Method to Compute Polygonal Minimal Surfaces, Preprint 33 SFB 288, Berlin 1992.
  • [Hil] S. Hildebrandt, Boundary Behaviour of Minimal Surfaces, Arch. Rat. Mech. Anal. 35 (1969), 47-82. MR 40:1901
  • [Hu] J.E. Hutchinson, Computing Conformal Maps and Minimal Surfaces, Proc. C.M.A., Canberra 26 (1991), 140-161. MR 92m:53012
  • [Ja] W. Jäger, Das Randverhalten von Flächen beschränkter mittlerer Krümmung bei $C^{1,\alpha}$-Rändern, Nachr. Akad. Wiss. Gött., II. Math. Phys. Kl. (1977), Nr. 5. MR 58:29204
  • [J] H. Jarausch, Zur numerischen Behandlung von parametrischen Minimalflächen mit Finite-Elementen, Dissertation Bochum 1978.
  • [JT] C. Johnson & V. Thomée, Error Estimates for a Finite Element Approximation of a Minimal Surface, Math. Comp. 29 (1975), 343-349. MR 53:4571
  • [N1] J.C.C. Nitsche, The Boundary Behaviour of Minimal Surfaces. Kellogg's Theorem and Branch Points on the Boundary, Invent. Math. 8 (1969), 313-333. MR 41:4399a
  • [N2] J.C.C. Nitsche, Lectures on Minimal Surfaces Volume 1, Cambridge University Press 1989. MR 90m:49031
  • [O] R. Osserman, A Proof of the Regularity Everywhere of the Classical Solution to Plateau's Problem, Ann. Math. 91 (1970), 550-569. MR 42:979
  • [P] H. Parks, Explicit Determination of Area Minimizing Hypersurfaces II, Mem. A.M.S. 60 (1986), no. 342 . MR 87h:49045
  • [PP] U. Pinkall, K. Polthier, Computing Discrete Minimal Surfaces and Their Conjugates, Exp. Math. 2 (1993), 15-36. MR 94j:53009
  • [R] T. Rado, On Plateau's Problem, Ann. Math. 2 (1930), 457-469.
  • [Ra] R. Rannacher, Some Asymptotic Error Estimates for Finite Element Approximation of a Minimal Surface, Rev. Française Automat. Informat. Recherche Opérationnelle Sér Rouge Anal. Numér. 11 (1977), 181-196. MR 56:4199
  • [St1] M. Struwe, On a Critical Point Theory for Minimal Surfaces Spanning a Wire, J. Reine Angew. Math 349 (1984), 1-23. MR 87a:58045
  • [St2] M. Struwe, Plateau's Problem and the Calculus of Variations, Mathematical Notes 35, Princeton University Press 1988. MR 90h:58016
  • [Ste] G. Steinmetz, Numerische Approximation von allgemeinen parametrischen Minimalflächen im $\mathbb{R}^3$, Forschungsarbeit FHS Regensburg 1987.
  • [Su] J. Sullivan, A Crystalline Approximation Theorem for Hypersurfaces, Princeton Ph.D. thesis 1990.
  • [To] F. Tomi, On the finite solvability of Plateau's problem, Lect. Notes Math. 597, Springer, 679-695, 1977. MR 56:13118
  • [T1] T. Tsuchiya, On Two Methods for Approximation Minimal Surfaces in Parametric Form, Math. Comp. 46 (1986), 517-529. MR 87d:49043
  • [T2] T. Tsuchiya, Discrete Solution of the Plateau Problem and its Convergence, Math. Comp. 49 (1987), 157-165. MR 88i:49032
  • [T3] T. Tsuchiya, A Note on Discrete Solutions of the Plateau Problem, Math. Comp. 54 (1990), 131-138. MR 91c:49063
  • [U] A. Underwood, Constructing Barriers to Minimal Surfaces from Polyhedral Data, Princeton Ph.D. thesis 1993.
  • [Wa1] H. J. Wagner, Ein Beitrag zur numerischen Approximation von Minimalflächen, Computing 19 (1977), 35-58.
  • [Wa2] H. J. Wagner, Consideration of Obstacles in the Numerical Approximation of Minimal Surfaces, Computing 19/4 (1978), 375-378. MR 58:3567
  • [Wi] W. L. Wilson jr., On Discrete Dirichlet and Plateau Problem, Num. Math. 3 (1961), 359-373. MR 25:761
  • [Wo] O. Wohlrab, Zur numerischen Behandlung von parametrischen Minimalflächen mit halbfreien Rändern, Dissertation Bonn 1985.

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Additional Information

Gerhard Dziuk
Affiliation: Institut für Angewandte Mathematik, Universität Freiburg, Hermann–Herder–Str. 10, D-79104 Freiburg i. Br., GERMANY
Email: gerd@mathematik.uni-freiburg.de

John E. Hutchinson
Affiliation: Department of Mathematics, School of Mathematical Sciences, Australian National University, GPO Box 4, Canberra, ACT 0200, AUSTRALIA
Email: John.Hutchinson@anu.edu.au

DOI: https://doi.org/10.1090/S0025-5718-99-01025-X
Keywords: minimal surface, finite elements, order of convergence, Plateau Problem
Received by editor(s): August 26, 1996
Article copyright: © Copyright 1999 American Mathematical Society

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