On two methods for approximating minimal surfaces in parametric form

Tsuchiya, Takuya

doi:10.1090/S0025-5718-1986-0829622-1

On two methods for approximating minimal surfaces in parametric form

Author: Takuya Tsuchiya
Journal: Math. Comp. 46 (1986), 517-529
MSC: Primary 49D20; Secondary 49F10, 65E05
DOI: https://doi.org/10.1090/S0025-5718-1986-0829622-1
MathSciNet review: 829622
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Two methods for approximating minimal surfaces in parametric form are considered. One minimizes the area of the surface, and the other the energy of the surface. The convergence of the algorithm of the first method is proved. The application of the second method to the approximation of conformal maps is examined. Several examples of computations are given.

[1] R. Courant, Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, Interscience Publishers, Inc., New York, N.Y., 1950. Appendix by M. Schiffer. MR 0036317
[2] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
[3] Masahiro Hinata, Masaaki Shimasaki, and Takeshi Kiyono, Numerical solution of Plateau’s problem by a finite element method, Math. Comp. 28 (1974), 45–60. MR 0331819, https://doi.org/10.1090/S0025-5718-1974-0331819-6
[4] H. Blaine Lawson Jr., Lectures on minimal submanifolds. Vol. I, Monografías de Matemática [Mathematical Monographs], vol. 14, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1977. MR 527121
H. Blaine Lawson Jr., Lectures on minimal submanifolds. Vol. I, 2nd ed., Mathematics Lecture Series, vol. 9, Publish or Perish, Inc., Wilmington, Del., 1980. MR 576752
[5] H. D. Mittelmann, On the approximation of capillary surfaces in a gravitational field, Computing 18 (1977), no. 2, 141–148 (English, with German summary). MR 0436779, https://doi.org/10.1007/BF02243623
[6] H. D. Mittelmann, On pointwise estimates for a finite element solution of nonlinear boundary value problems, SIAM J. Numer. Anal. 14 (1977), no. 4, 773–778. MR 0445865, https://doi.org/10.1137/0714053
[7] Hans-Detlef Mittelmann, On the approximate solution of nonlinear variational inequalities, Numer. Math. 29 (1977/78), no. 4, 451–462. MR 0658146, https://doi.org/10.1007/BF01432881
[8] Robert Osserman, A survey of minimal surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. MR 0256278