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

Abstract

A distributed elliptic control problem with control constraints is considered, which is formulated as 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 control is associated with the residual of the variational inequality but does not appear in the weak formulation. Each of the three variables is discretized independently by hp-finite elements. In particular, the non-penetration condition of the control variable is relaxed to a finite set of quadrature 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 the mesh size as well as of the polynomial degree. Moreover, reliable and efficient a posteriori error estimates are presented, which enable hp-adaptive mesh refinements. Several numerical experiments demonstrate the applicability of the discretization with hp-finite elements, the efficiency of the a posteriori error estimates and the improvements with respect to the convergence order resulting from the application of hp-adaptivity. In particular, the hp-adaptive schemes lead to superior convergence properties.