Numerical solution of Variational Inequalities by Adaptive Finite Elements

Franz-Theo Suttmeier describes a general approach to a posteriori error estimationand adaptive mesh design for finite element models where the solutionis subjected to inequality constraints. This is an extension to variationalinequalities of the so-called Dual-Weighted-Residual method (DWR method)which is based on a variational formulation of the problem and uses globalduality arguments for deriving weighted a posteriori error estimates with respectto arbitrary functionals of the error. In these estimates local residuals ofthe computed solution are multiplied by sensitivity factors which are obtainedfrom a numerically computed dual solution. The resulting local error indicatorsare used in a feed-back process for generating economical meshes whichare tailored according to the particular goal of the computation. This methodis developed here for several model problems. Based on these examples, a generalconcept is proposed, which provides a systematic way of adaptive errorcontrol for problems stated in form of variational inequalities.