The aim of this paper is the study of the convergence of a finite element approximation for a variational inequality related to free boundary problems in non-steady fluid flow through porous media. There have been many results in the stationary case, for example, the steady dam problems, the steady flow well problems, etc. In this paper we shall deal with the axisymmetric non-steady porous flow well problem. It is well know that by means of Torelli's transform this problem, similar to the non-steady rectangular dam problem, can be reduced a variational, inequality, and the existence, uniqueness and regularity of the solution can be obtained ([12, 7]). Now we study the numerical solution of this variational inequality. The main results are as follows:
1. We establish new regularity properties for the solution $W$ of the variation inequality. We prove that $W \in L^\infty(0, T; H^2(D))$, $γ_0W\in L^\infty(0, T; H^2(T_n))$ and $D_1γ_0W\in L^2(0, T; H^1(T_n))$ (see Theorem 2.5). Friedman and Torelli [7] obtained $W\in L^2(0, T; H^2(D))$. Our new regularity properties will be used for error estimation.
2. We prove that the error estimate for the finite element solution of the variational inequality is $$ ( \sum^N_{i=1}\| W^1 - W^1_h \|^2_{H^1(D)}\Delta t)^{1/2} = O(h+\Delta t^{1/2})$$ (see Theorem 3.4). In the stationary case the error estimate is $\|W-W_h\|_{H^1(D)} = O(k)$ ([3,6]).
3. We give a numerical example and compare the result with the corresponding result in the stationary case.
The result of this paper are valid for the non-ready rectangular dam problem with stationary or quasi-stationary initial data (see [7], p.534).