Keywords: plateau problem, variational form, convexity, brouwer's fixed point theorem, finite element, maximum value principle, multigrid method.

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 well-known 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.

1

K. Brakke, "The surface evolver. Experiment", Math. 1, no. 2, 141-165, 1992.

2

D. Hoffman, "Natural minimal surfaces. Via theory and computation", Science Television, NewYork; American Mathematical Society, Providence, RI, 1990.

3

H. Gu, "The Software Generating of Minamal Surface Subject to the Plateau Problems", PhD thesis in process, RISC-Linz, 2002.

4

C. Johnson and V. Thomeé, "Error estimates for a finite element approximation of a minimal surface", Math. of Comp. Vol. 29, Num. 130, 1975 343-349. doi:10.2307/2005555

5

J. Xu and A. Zhou, "Local and parallel finite element algorithms based on two-grid discretizations", Math. Comp., 69(2000), 881-909. doi:10.1090/S0025-5718-99-01149-7

6

J. Xu and A. Zhou, "Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems", Adv. Comput. Math., 14(2001), 393-327. doi:10.1023/A:1012284322811

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £125 +P&P)