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CivilComp Proceedings
ISSN 17593433 CCP: 75
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and Z. Bittnar
Paper 19
The Finite Element Approximation for Minimal Surfaces Subject to the Plateau Problems H. Gu
RISCLinz Institute, Johannes Kepler University, Linz, Austria H. Gu, "The Finite Element Approximation for Minimal Surfaces Subject to the Plateau Problems", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 19, 2002. doi:10.4203/ccp.75.19
Keywords: plateau problem, variational form, convexity, brouwer's fixed point theorem, finite element, maximum value principle, multigrid method.
Summary
A variety of software for generating the minimal
surfaces of special types [1,2] are already available.
However, the convergence theories
for those approximating methods are always left uncompleted. This leads to the
difficulty of automatical displaying a whole class of minimal surfaces
represented in a general form.
In this paper, we discuss the finite element
methods for solving the minimal surfaces subject to the
wellknown Plateau probems. This work, which belongs to the aspects
of one research project [3], has been partially supported
by the Austrian "Fonds zur Förderung der wissenschaftlichen Forschung
(FWF)" under project nr. SFB F013/F1304.
From the context of this paper, the boundary value differential form for the plateau problem is given at first and we could generate its associated discrete scheme in variational forms by the finite element method. For solving the numerical solution, either the Newton's iteration or other approximation approaches can be applied on, or we can try some promising symbolic methods. The convergence properties of the numerical solutions are proved based on different kind of finite element spaces and, some multigrid algorithms have also been proposed and implemented to speed up the computation. The possibility of applying the parallel algorithms [5,6] which are well suited to the solution of problems on the large scale domains. Compared to other existing finite element apporaches [4] for solving Plateau problems, the discrete forms proposed in the paper have much lower complexity and we have already started to apply these new approximation methods for generating the minimal surface graphically on certain softwares. References
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