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

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 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**, 10.1090/S0025-5718-1974-0331819-6**[4]**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****[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****[7]**Hans-Detlef Mittelmann,*On the approximate solution of nonlinear variational inequalities*, Numer. Math.**29**(1977/78), no. 4, 451–462. MR**0658146****[8]**Robert Osserman,*A survey of minimal surfaces*, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. MR**0256278**

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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