hp-finite elements for elliptic optimal control problems with control constraints

A distributed elliptic control problem with control constraints is considered, which is formulatedas a three field problem and consists of two variational equations for the state and the co-state variables as well as of a variational inequality for the control variable. The adjoint controlis associated with the residual of the variational inequality but does not appear in the weakformulation. Each of the three variables is discretized independently by ℎ??-finite elements. Inparticular, the non-penetration condition of the control variable is relaxed to a finite set ofquadrature points. Sufficient conditions for the unique existence of a discrete solution are stated.Also a priori error estimates and guaranteed convergence rates are derived in terms of themesh size as well as of the polynomial degree. Moreover, reliable and efficient a posteriorierror estimates are presented, which enable ℎ??-adaptive mesh refinements. Several numericalexperiments demonstrate the applicability of the discretization with ℎ??-finite elements, theefficiency of the a posteriori error estimates and the improvements with respect to the convergenceorder resulting from the application of ℎ??-adaptivity. In particular, the ℎ??-adaptive schemes leadto superior convergence properties.