A Variational Inequality Approach for One Dimensional Stefan Problem

In this paper, we develop a numerical method to solve a famous free boundary PDE called the one dimensional Stefan problem. First, we rewrite the PDE as a variational inequality problem (VIP). Using the linear finite element method, we discretize the variational inequality and achieve a linear complementarity problem (LCP). We present some existence and uniqueness theorems for solutions of the variational inequalities and free boundary problems. Finally we solve the LCP numerically by applying a modification of the active set strategy.