Finite Element Approximations to One-Phase Nonlinear Free Boundary Problem in Groundwater Contamination Flow

In this article, we consider a single-phase coupled nonlinear Stefan problem of the water-head and concentration equations with nonlinear source and permeance terms and a Dirichlet boundary condition depending on the free-boundary function. The problem is very important in subsurface contaminant transport and remediation, seawater intrusion and control, and many other applications. While a Landau type transformation is introduced to immobilize the free boundary, a transformation for the water-head and concentration functions is defined to deal with the nonhomogeneous Dirichlet boundary condition, which depends on the free boundary function. An H 1-finite element method for the problem is then proposed and analyzed. The existence of the approximation solution is established, and error estimates are obtained for both the semi-discrete schemes and the fully discrete schemes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 22: 1267–1288, 2006