Semismooth Newton methods for variational problems with inequality constraints

Inequality constraints occur in many different fields of application, e.g., in structural mechanics, flow processes in porous media or mathematical finance. In this paper, we make use of the mathematical structure of these conditions in order to obtain an abstract computational framework for problems with inequality conditions. The constraints are enforced locally by means of Lagrange multipliers which are defined with respect to dual basis functions. The reformulation of the inequality conditions in terms of nonlinear complementarity functions leads to a system of semismooth nonlinear equations that is solved by a generalized version of Newton's method for semismooth problems. By this, both nonlinearities in the pde model and inequality constraints are treated within a single Newton iteration which converges locally superlinear. The scheme can efficiently be implemented in terms of an active set strategy with local static condensation of the non-essential variables. Numerical examples from different fields of application illustrate the generality and the robustness of the method.