A wide range of free boundary problems occurring in engineering and industry can be rewritten as a minimization problem for a strictly convex, piecewise smooth but non--differentiable energy functional, or even more, as a variational inequality. The algebraic solution of the related discretized problem is a very delicate question, because usual Newton techniques cannot be applied.
We propose a new approach based on convex minimization and constrained Newton type linearization. While convex minimization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the convergence speed. We present a general convergence theory and discuss several applications