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The article proposes a nonoverlapping domain decomposition method for variational inequalities derived from free boundary problems. The free boundary value problem is broken up into two problems on nonoverlapping regions. In one region the problem is treated as a partial differential equation, while in the second region that contains the free boundary part, a variational inequality is considered. By solving these two related problems successively, we have shown that the successive solutions converge to the solution of the original problem. Application to a free surface seepage problem is given. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

The nonlinear equation describing the unsteady and unsaturated, one-dimensional vertical flow in porous media is formulated with the finite element representation employing a Galerkin approach.

Summary Seepage through porous media in most instances is not a two-dimensional flow phenomenon. One such situation which is investigated herein is the flow from a pond with a bottom formed by the rotation of a curve consisting of a small segment being horizontal and the remainder being an arbitrary convex shaped curve. The approach used to solve this free surface axisymmetric seepage problem is an alternating iteration scheme in conjunction with the Baiocchi transformation and method. The problem is split into two overlapping regions, one in which the free surface is treated and the other in which the singularity is treated. This approach does not require moving meshes and allows a very general prescribed shape for the pond boundary. The numerical approach to the reformulated seepage problem is presented along with several numerical results and comparisons. Also presented is a proof of uniqueness of the solution for such problems provided we make a certain smoothness assumption for the free surface.

A finite element, weighted residual, successive substitution algorithm has been used to solve two-dimensional infiltration problems whose governing partial differential equation and boundary conditions are highly nonlinear. The results compared favorably with other published numerical solutions to similar examples. The soil water diffusivity, as well as the capillary conductivity, were approximated within a finite element in terms of shape functions and discretized values of these functions calculated using guessed values or previously calculated values at a certain time level of the water content at the nodes. This successive substitution scheme converged rapidly within each time step. The method is general and can be extended to three-dimensional flows and the use of higher order approximations.

Summary Free boundary value problems, too complicated for formulation as a variational inequality, are broken up into two problems on overlapping regions. On one region the problem is treated as an ordinary boundary value problem; on the second region, the “free boundary part” of the problem is reduced to a variational inequality. By solving the two problems successively it is shown that under certain conditions the successive solutions converge to a single function that gives a solution of the original problem. Application to a filtration problem is given.

An experiment and a numerical scheme which combine previously given methods for the dispersion and seepage phenomena have been presented to describe the dispersion of a pollutant in porous media beneath a triangular channel. The combined effect of longitudinal and lateral dispersion was considered. Numerical concentration profiles are given along with a comparison of experimental and numerical data. The technique suggested herein can be extended to other channel shapes and flow fields accounting for the capillary fringe.