A stable nonconforming mixed finite element scheme for elliptic optimal control problems

A new stable nonconforming mixed finite element (MFE) scheme is proposed for the constrained optimal control problem (OCP) governed by elliptic equations. The nonconforming FEs are employed to approximate the co-state, the piecewise constant element for the state and control, respectively. The corresponding optimal order error estimates are derived by full use of the distinguish properties of the nonconforming element, i.e., (a) the divergence of the function in the FE space is a constant; (b) the consistency error is one order higher than its interpolation error in the broken energy norm. Finally, some numerical results are carried out to verify the theoretical analysis.