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

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 prob- lem 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. Introduction. Methods for the numerical solution of the Plateau problem have so far been examined by D. Greenspan (3), (4), using the combination technique of difference and variational methods, and by P. Concus (5), using a finite difference method. These two methods can be applied only to the so-called restricted form of the Plateau problem described by Forsythe and Wasow (2, Section 18.9), that is, to the problems where the boundary condition is represented by a single-valued function. Thus, they cannot be applied to the problem where the boundary condition is repre- sented by a multi-valued function, such as Courant's example described later. This paper shows that such multi-valued boundary-value problems can be solved numerically by a finite element method. In this case, two solution methods, one for a free boundary problem and the other in a cyclindrical coordinate system, are presented.