Mesh adaptivity in optimal control of elliptic variational inequalities with point-tracking of the state

Abstract

An adaptive finite element method is developed for a class of optimal control problems with elliptic variational inequality constraints and objective functionals defined on the space of continuous functions, necessitated by a point-tracking requirement with respect to the state variable. A suitable first order stationarity concept is derived for the problem class via a penalty technique. The dual-weighted residual approach for goal-oriented adaptive finite elements is applied and relies on the stationarity system. It yields primal residuals weighted by approximate dual quantities and vice versa as well as complementarity mismatch errors. A report on numerical tests, including the critical case of biactivity, completes this work.