Adaptive finite element methods for variational inequalities

Abstract

In this paper we are concerned with the numerical solution of stationary variational inequalities of obstacle type associated with second order elliptic differential operators in two or three space dimensions. In particular, we present adaptive finite element techniques featuring multilevel iterative solvers and a posteriori error estimators for local refinement of the triangulations. The algorithms rely on an outer-inner iterative scheme with an outer active set strategy and inner multilevel preconditioned cg-iterations involving variants of the hierarchical and the BPX-preconditioner which are derivded in the framework of multilevel additive Schwarz iterations. For the a posteriori error estimation in the energy norm three error estimators are presented which are based on the approximate solution of a quasivariational inequality satis- fied by a piecewise quadratic approximation of the global discretization error. Finally, the performance of the preconditioners and the error estimators is illustrated by numerical results for a wide variety of stationary free boundary problems.