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

Pan Xue; Xiaobin Ye; Zhifeng Weng; Pan Xue; Xiaobin Ye; Zhifeng Weng

doi:10.3934/era.2026261

Electronic Research Archive

2026, Volume 34, Issue 9: 5887-5906. doi: 10.3934/era.2026261

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

Fujian Province University Key Laboratory of Computation Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

Received: 16 April 2026 Revised: 29 June 2026 Accepted: 03 July 2026 Published: 13 July 2026

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.
Citation: Pan Xue, Xiaobin Ye, Zhifeng Weng. A lowest equal-order stabilized mixed finite element method for an elliptic optimal control problem with control constraints[J]. Electronic Research Archive, 2026, 34(9): 5887-5906. doi: 10.3934/era.2026261

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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.

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