A survey of free boundary value problems in the theory of fluid flow through porous media: variational inequality approach - Part I

A survey is presented of work done using variational inequalities in solving free-boundary problems arising in the filtration of liquids through porous materials. In 1971 Baiocchi proposed this new method for solving such problems. This method proved effective not only from the purely theoretical point of view, that is, for proving the existence and uniqueness of a solution, but also from the point of view of yielding new, simple and efficient numerical solution schemes. This new method was systematically and extensively developed by mathematicians at the Laboratory of Numerical Analysis at the National Research Centre in collaboration with the Institute of Hydraulics of the University of Pavia. The aim of this survey is to present the basic ideas of this method, its applications, and other problems to which various researchers have extended the method. Also, the paper is written in such a manner as to be understood by a more general audience other than mathematicians, who are the ones who have done the majority of work in this area.