A Nonconforming Finite Element Method for Constrained Optimal Control Problems Governed by Parabolic Equations

Abstract

In this paper, a nonconforming finite element method (NFEM) is proposed for the constrained optimal control problems (OCPs) governed by parabolic equations. The time discretization is based on the finite difference methods. The state and co-state variables are approximated by the nonconforming $EQ_1^{\operatorname{rot}}$ elements, and the control variable is approximated by the piecewise constant element, respectively. Some superclose properties are obtained for the above three variables. Moreover, for the state and co-state, the convergence and superconvergence results are achieved in $L^2$-norm and the broken energy norm, respectively.