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

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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, New York, 1950. MR**0036317 (12:90a)****[2]**R. Courant & D. Hilbert,*Methods of Mathematical Physics*, vol. II, Interscience, New York, 1962. MR**0065391 (16:426a)****[3]**M. Hinata, M. Shimasaki & T. Kiyono, "Numerical solution of Plateau's problem by a finite element method,"*Math. Comp.*, v. 28, 1974, pp. 45-60. MR**0331819 (48:10151)****[4]**H. B. Lawson,*Lectures on Minimal Submanifolds*, Vol. I, Publish or Perish, Berkeley, 1980. MR**576752 (82d:53035b)****[5]**H. D. Mittelmann, "On the approximation of capillary surface in a gravitational field,"*Computing*, v. 18, 1977, pp. 141-148. MR**0436779 (55:9718)****[6]**H. D. Mittelmann, "On the pointwise estimates for a finite element solution of nonlinear boundary value problems,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 773-778. MR**0445865 (56:4198)****[7]**H. D. Mittelmann, "On the approximate solution of nonlinear variational inequalities,"*Numer. Math.*, v. 29, 1978, pp. 451-462. MR**0658146 (58:31922)****[8]**R. Osserman,*A Survey of Minimal Surfaces*, Van Nostrand Reinhold, New York, 1969. MR**0256278 (41:934)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0829622-1

Keywords:
Minimal surface,
parametric form,
parametrization,
conformal map,
relaxation method,
the Douglas-Rado solution of the classical Plateau problem

Article copyright:
© Copyright 1986
American Mathematical Society