On the numerical approximation and computation of minimal surface continua bounded by one-parameter families of polygonal contours

Applied Numerical Mathematics

Volumn 25, Issue 1, 1997, Pages 89-116

  Hinze, Michael ^a  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; BIFURCATION (MATHEMATICS); FINITE ELEMENT METHOD; NUMERICAL METHODS; PROBLEM SOLVING; SURFACES;

FINITE DIMENSIONAL SPACES; JORDAN CURVE; MINIMAL SURFACE CONTINUA; POLYGONAL CONTOURS;

COMPUTATIONAL METHODS;

EID: 0031257053     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(97)00045-7     Document Type: Article

References (34)

1. M.J. Beeson and A.J. Tromba, The cusp catastrophe of thom in the bifurcation of minimal surfaces, Manuscripta Math. 46 (1984) 273-308.
2. K.A. Brakke, The surface evolver, Experiment. Math. 1 (1992) 141-165.
3. K.A. Brakke, The surface evolver and the stability of liquid surfaces, Philos. Trans. Roy. Soc. London Ser. A 354 (1996) 2143-2157.
4. J. Büch, Bifurkation von Minimalflächen und elementare Katastrophen, Manuscripta Math. 55 (1986) 269-306.
5. R. Courant, Dirichlet's Principle (Interscience Publishers, 1950).
6. P. Deuflhard, B. Fiedler and P. Kunkel, Efficient numerical pathfollowing beyond critical points, SIAM J. Numer. Anal. 24 (1987) 912-927.
7. J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263-321.
8. G. Dziuk, An algorithm for evolutionary surfaces, Numer. Math. 58 (1991) 603-611.
9. G. Dziuk and J.E. Hutchinson, On the approximation of unstable parametric minimal surfaces, Preprint No. 2o. 340, SFB 256, Universität Bonn, Germany (1994).
10. E. Heinz, Über die analytische Abhängigkeit eines linearen elliptischen Randwertproblems von Parametern, Nachr. Akad. Wiss. Göttingen Math.-Phys. K1. II (1979) 1-20.
11. E. Heinz, Über eine Verallgemeinerung des Plateau'schen Problems, Manuscripta Math. 28 (1979) 81-88.
12. E. Heinz, Ein mit der Theorie der Minimalflächen zusammenhängendes Variationsproblem, Nachr. Akad. Wiss. Göttingen Math.-Phys. K1. II (1980) 25-35.
13. E. Heinz, Zum Marx-Shiffmanschen Variationsproblem, J. Reine Angew. Math. 344 (1983) 196-200.
14. E. Heinz, Minimalflächen mit polygonalem Rand, Math. Z. 183 (1983) 547-564.
15. M. Hinze, Zur numerischen Behandlung des Marx-Shiffman-Randwertproblems, Diplomarbeit, Institut für Angew. Math., Universtät Bonn (1989).
16. M. Hinze, On the numerical treatment of quasiminimal surfaces, Impact Comput. Sci. Engrg. 5 (1993) 249-270.
17. M. Hinze, On the numerical treatment of minimal surfaces with polygonal boundaries, Ph.D., Technical University Berlin, Germany (1994).
18. M. Hinze, On the numerical approximation of unstable minimal surfaces with polygonal boundaries, Numer. Math. 73 (1996) 95-118.
19. J.E. Hutchinson, Computing conformal maps and minimal surfaces, Proc. Centre Math. Anal. Austral. Nat. Univ. 26 (1991).
20. H. Jarausch, Zur numerischen Behandlung von parametrischen Minimalflächen mit Finiten Elementen, Dissertation, Ruhr Universität Bochum, Germany.
21. F. Lewerenz, Eine Bemerkung zu den Marx-Shiffman'schen Minimalvektoren bei Polygonen, Arch. Rational Mech. Anal. 78 (1981).
22. I. Marx, On the classification of unstable minimal surfaces with polygonal boundaries, Comm. Pure Appl. Math. 8 (1955) 235-244.
23. J.C.C. Nitsche, Lectures on Minimal Surfaces, Vol. 1 (Cambridge University Press, 1989).
24. J.C.C. Nitsche, Contours bounding at least three solutions of Plateau's problem, Arch. Rational Mech. Anal. 30 (1968) 1-11.
25. U. Pinkall and K. Polthier, Computing discrete minimal surfaces and their conjugates, Preprint No. 49, SFB 288, Technical University Berlin, Germany.
26. J. Pousin and J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems, Numer. Math. 69 (1994) 213-231.
27. H. Ruchert, A uniqueness result for Enneper's minimal surface, Indiana Univ. Math. J. 30 (1981) 427-431.
28. p mit polygonalem Rand, Math. Z. 189 (1985) 167-184.
29. p mit polygonalem Rand, Reine Angew. Math. 358 (1985) 92-96.
30. p with polygonal boundaries, Manuscripta Math. 53 (1985) 167-197.
31. T. Tsuchija, On two methods for approximating minimal surfaces in parametric form, Math. Comp. 46 (1986) 512-529.
32. T. Tsuchija, Discrete solution of the Plateau problem and its convergence, Math. Comp. 49 (1987) 157-165.
33. T. Tsuchija, A note on discrete solutions of the Plateau problem, Math. Comp. 54 (1990) 131-138.
34. O. Wohlrab, Zur numerischen Behandlung von parametrischen Minimalflächen mit halbfreien Rändern, Preprint No. 739, SFB 72, Universität Bonn.