Monotone Iterations for Elliptic Variational Inequalities

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. The fast solution of related discretized problems 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 min- imization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the conver- gence speed. We present a general convergence theory and discuss several applications.