A nonconforming finite element method (FEM) is proposed for optimal control problems (OCPs) governed by monotone semilinear elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection of the adjoint state, respectively. Some global supercloseness and superconvergence estimates are achieved by making full use of the distinguish characters of this element, such as the interpolation equals to its Ritz projection, and the consistency error is 1 − ε ( is small enough) order higher than its interpolation error in the broken energy norm when the exact solution belongs to H^3 − ε(Ω). Finally, some numerical results are presented to verify the theoretical analysis.

中文翻译：

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单调半线性椭圆最优控制问题的非协调有限元方法的超收敛性分析

针对单调半线性椭圆方程的最优控制问题，提出了一种非协调有限元方法。状态和伴随状态分别由不合格元素逼近，而控制分别由伴随状态的正交投影逼近。通过充分利用该元素的不同特征，可以实现一些全局超闭合性和超收敛估计，例如，插值等于其Ritz投影，并且一致性误差比其插值误差高1-  ε（足够小）。精确解属于H ^{3-  ε}时的破坏能量范数（Ω）。最后，给出一些数值结果以验证理论分析。