Generating Minimal Surfaces Subject to the Plateau Problems by Finite Element Method

摘要

There already exists avariety of softwares for generating minimal surfaces of special types. However, the convergence theories for those approximating methods are always left uncompleted. This leads to the difficulty of displaying a whole class of minimal surface in general form. In this paper, we discuss the finite element approximating methods to the minimal surfaces which are subject to the well-known Plateau probems. The differential form of the Plateau problems will be given and, for solving the associated discrete scheme, either the numerical Newton iteration method can be applied or we can try some promsing symbolic approaches. The convergence property of the numerical solutions is proved and this method will be applied to generating the minimal surface graphically on certain softwares later. The method proposed in this paper has much lower complexity and fits for inplementing the two grid and parallel algorithms to speed up the computation.