Abstract

Motivated by the obstacle problem as well as by optimization problems with partial differential equations subject to pointwise constraints on the control, the state or its derivative, semismooth Newton methods and Moreau-Yosida based path-following techniques will be discussed. Besides the convergence analysis in function space, mesh independence properties of the iterations are presented and numerical analysis aspects, such as the optimal link between the Moreau-Yosida parameter and the mesh-width of discretization as well as adaptive finite element methods, will be addressed. Quasi-variational inequalities involving the $p$-Laplacian and constraints on the gradient of the state will be briefly studied, too. Finally, the potential of the introduced methodology is highlighted by means of various applications ranging from phase-separation processes to problems in mathematical image processing.