A lowest equal-order stabilized mixed finite element method for an elliptic optimal control problem with control constraints

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Abstract

We propose a lowest equal-order stabilized mixed finite element method for an elliptic optimal control problem with box control constraints. First, optimality conditions are derived based on the variational principle, transforming the original problem into a coupled system consisting of the state equation, the costate equation, and a variational inequality. The state and costate variables are approximated using the lowest equal-order mixed finite element pair, and the control variable is approximated by piecewise constant functions. A stabilization term for velocity projection based on the difference between two local Gauss integrations is introduced to circumvent the Ladyzhenskaya– Babuška–Brezzi (LBB) condition. Subsequently, a priori error estimates for the state, costate, and control variables are derived. Finally, 2D and 3D numerical experiments are performed to validate the theoretical analysis and the efficiency of the method.

Keywords

elliptic optimal control problem / box control constraints / $P_{1}^{2}-P_{1}$ mixed finite element / velocity projection stabilization method / a priori error estimates

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