Variational formulation of a model free-boundary problem

The purpose of this work is to present an error analysis of the numerical approximation by a finite element method of a free-surface problem. The analysis has been done in an abstract model which has many of the features of a free-surface problem for a viscous liquid. We study in this paper how the numerical approximation of the free boundary affects the approximation of the other variables of the problem and vice versa. We present the numerical analysis of a free-boundary problem that is intended to incorporate many of the difficulties found in a class of models of fluid-flow phenomena with free surfaces. One such phenomenon which motivates the current work is the flow of a liquid constrained only partly by a container, that is, in which a part of the boundary of the domain filled by the liquid is an interface with another liquid of much smaller density, and for which surface tension plays a significant role in determining the shape of the free surface. One model for the behavior of such liquids is based on the assumption that the surface tension between the two liquids is proportional to the curvature of the free surface; the constant of proportionality is a physical property of the two fluids. This model has been studied extensively in recently years, both experimentally (cf. Jean and Pritchard [15] and Pritchard [19]), theoretically (cf. Allain [2], Beale [4], Bemelmans [5], Jean [14], Pukhnachov [20], and Solonnikov [27]), asymptotically (Keller and Miksis [16]) and computationally (cf. Cuvelier [10], Ryskin and Leal [23], and Saito and Scriven [24]). Our purpose here is to establish a framework for the analysis of convergence properties of the computational techniques being used. The only previous work that we are aware of in this direction is by Nitsche [18]. In the first section of the paper, we define our model problem in classical terms. In the second section, we construct a variational formulation for the problem that has two new features. One is that it allows the existence of a solution to be proved with weaker assumptions on the data than has been possible Received February 26, 1990; revised November 20, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 65N30.

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