A finite element method for a unidimensional single-phase nonlinear free boundary problem in groundwater flow

Consider a unidimensional, single-phase nonlinear Stefan problem with nonlinear source and permeance terms, and a Dirichlet boundary condition depending on the free boundary function. This problem is important in groundwater flow. By immobilizing the free boundary with the help of a Landau-type transformation, together with a homogeneous transformation dealing with the nonhomogeneous Dirichlet boundary condition, an H1-finite element method for the problem is proposed and analyzed. Global existence of the approximate solution is established, and optimal error estimates in L2, L∞, H1 and H2 norms are derived for both semi-discrete and fully discrete schemes.