Numerical solution of Plateau's problem by a finite element method

Authors:
Masahiro Hinata, Masaaki Shimasaki and Takeshi Kiyono

Journal:
Math. Comp. **28** (1974), 45-60

MSC:
Primary 65N35; Secondary 49F10

MathSciNet review:
0331819

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Abstract: This paper concerns the application of a finite element method to the numerical solution of a nonrestricted form of the Plateau problem, as well as to a free boundary problem of Plateau type. The solutions obtained here are examined for several examples and are considered to be sufficiently accurate.

It is also observed that the hysteresis effect, which is a feature of a nonlinear problem, appears in this problem.

**[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]**George E. Forsythe and Wolfgang R. Wasow,*Finite-difference methods for partial differential equations*, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. MR**0130124****[3]**Donald Greenspan,*On approximating extremals of functionals. I. The method and examples for boundary value problems*, ICC Bull.**4**(1965), 99–120. MR**0191117****[4]**D. Greenspan,*On approximating extremals of functionals. II. Theory and generalizations related to boundary value problems for nonlinear differential equations*, Internat. J. Engrg. Sci.**5**(1967), 571–588 (English, with French, German, Italian and Russian summaries). MR**0220448****[5]**Paul Concus,*Numerical solution of the minimal surface equation*, Math. Comp.**21**(1967), 340–350. MR**0229394**, 10.1090/S0025-5718-1967-0229394-6**[6]**Robert S. Stepleman,*Finite-dimensional analogues of variational problems in the plane*, SIAM J. Numer. Anal.**8**(1971), 11–23. MR**0290539**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0331819-6

Keywords:
A boundary-value problem,
a finite element method,
a free boundary problem,
cylindrical coordinate system,
R. Courant's example,
the hysteresis effect of Plateau's problem

Article copyright:
© Copyright 1974
American Mathematical Society