In this paper, a nonconforming finite element method (FEM) is proposed for the Neumann type boundary optimal control problems (OCPs) governed by elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection through the adjoint state. Some superclose behaviors are derived by full use of the distinguish characters of this element, such as the interpolation operator equals to the Ritz projection, and the consistency error is higher than its interpolation error in the broken energy norm. After that, the global superconvergence results are obtained by employing the so-called post-interpolation technique. Finally, some numerical results are provided to verify the theoretical analysis.

中文翻译：

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具有椭圆方程的 Neumann 边界 OCP 的非一致 FEM 的全局超收敛分析

在本文中，针对由椭圆方程控制的 Neumann 型边界最优控制问题 (OCP)，提出了一种非一致性有限元方法 (FEM)。状态和伴随状态由不合格元素近似，控制由通过伴随状态的正交投影近似。一些超接近行为是充分利用该元素的区分特性推导出来的，例如插值算子等于Ritz投影，在破能量范数中一致性误差高于其插值误差。之后，采用所谓的后插值技术获得全局超收敛结果。最后，提供了一些数值结果来验证理论分析。