Nonconforming Finite Element Methods for the Constrained Optimal Control Problems Governed by Nonsmooth Elliptic Equations

In this paper, nonconforming finite element methods (FEMs) are proposed for the constrained optimal control problems (OCPs) governed by the nonsmooth elliptic equations, in which the popular EQ1rot element is employed to approximate the state and adjoint state, and the piecewise constant element is used to approximate the control. Firstly, the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem. Secondly, the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate. Lastly, the methods are extended to some other well-known nonconforming elements.