From inexact active set strategies to nonlinear multigrid methods

Due to their efficiency and robustness, linear multigrid methods lend themselves to be a starting point for the development of nonlinear iterative strategies for the solution of nonlinear contact problems, see, e.g., [3, 1, 10, 5]. One nonlinear strategy is to reduce the contact problem to a sequence of linear problems and to solve each of these by a linear multigrid method. This approach is often connected to active set strategies or semismooth Newton methods [6]. To avoid solving the linear problems exactly, one can use inexact active set strategies, see [7, 8]. The convergence of this inexact strategy depends on the accuracy the inner problem is solved with, see [7], as well as on algorithmic parameters [8]. A second strategy is to deal directly with the nonlinearity within the multigrid method by using, e.g., nonlinear smoothers and nonlinear interpolation operators, see [10, 9, 1]. Using the convex energy for controlling the iteration process, globally convergent nonlinear multigrid methods can be constructed which allow for solving contact problems with the speed of a linear multigrid method [10]. A third possibility is to employ a saddle point approach [3] and to solve for the primal and dual variables simultaneously using an algebraic multigrid method.