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Finite element approximation of $H$-surfaces

Authors: Yuki Matsuzawa, Takashi Suzuki and Takuya Tsuchiya
Journal: Math. Comp. 72 (2003), 607-617
MSC (2000): Primary 65N30, 35J65
Published electronically: October 22, 2002
MathSciNet review: 1954958
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Abstract: In this paper a piecewise linear finite element approximation of $H$-surfaces, or surfaces with constant mean curvature, spanned by a given Jordan curve in $\textbf{R}^3$ is considered. It is proved that the finite element $H$-surfaces converge to the exact $H$-surfaces under the condition that the Jordan curve is rectifiable. Several numerical examples are given.

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  • 1. H. Brezis and J.-M. Coron, Multiple solutions of $H$-systems and Rellich's conjecture, Comm. Pure Appl. Math. 37 (1984) 149-187. MR 85i:53010
  • 2. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. MR 58:25001
  • 3. P.G. Ciarlet, Basic Error Estimates for Elliptic Problems, In; P.G. Ciarlet and J.L. Lions (ed.), Finite Element Methods (Part 1), Handbook of Numerical Analysis, 17-351, Elsevier Science Publishers B.V., Amsterdam, 1991. MR 92f:65001
  • 4. R. Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience, New York, 1950. MR 12:90a
  • 5. U. Dierkes, S. Hildebrandt, A. Küster, O. Wohlrab, Minimal Surfaces I, Springer, 1992. MR 94c:49001a
  • 6. K. Grosse-Brauckmann and K. Polthier, Numerical examples of compact surfaces of constant mean curvature, In; Elliptic and Parabolic Methods in Geometry, A.K. Peters, 1996, 23-46. MR 97j:53008
  • 7. D.E. Hewgill, Computing surfaces of constant mean curvature with singularities, Computing, 32 (1984) 81-92. MR 85k:65089
  • 8. S. Hildebrandt, Über Flächen konstanter mittlerer Krümmung, Math. Z. 112 (1969) 107-144. MR 40:3446
  • 9. S. Hildebrandt, On the Plateau problem for surfaces of constant mean curvature, Comm. Pure Appl. Math. 23 (1970) 97-114. MR 41:932
  • 10. R. Osserman, A proof of the regularity everywhere of the classical solution to Plateau's problem, Ann. Math. 91 (1970) 550-569. MR 42:979
  • 11. A.H. Schatz, A weak discrete maximum principle and stability of the finite element method in $L^\infty$ on polygonal domains, Math. Comp. 34 (1980) 77-91. MR 81e:65063
  • 12. T. Tsuchiya, On two methods for approximating minimal surfaces in parametric form, Math. Comp. 46 (1986) 517-529. MR 87d:49043
  • 13. T. Tsuchiya, Discrete solution of the Plateau problem and its convergence, Math. Comp. 49 (1987) 157-165. MR 88i:49032
  • 14. T. Tsuchiya, A note on discrete solutions of the Plateau problem, Math. Comp. 54 (1990) 131-138. MR 91c:49063
  • 15. T. Tsuchiya, Finite element approximations of conformal mappings, Numer. Func. Anal. Optim. 22 (2001), 419-440.

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Additional Information

Takashi Suzuki
Affiliation: Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka 560-0043, Japan

Takuya Tsuchiya
Affiliation: Department of Mathematical Sciences, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan

Keywords: Finite element method, constant mean curvature, $H$-surfaces.
Received by editor(s): July 10, 2000
Received by editor(s) in revised form: February 28, 2001
Published electronically: October 22, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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