In this paper, nonconforming finite element methods (FEMs) are proposed for the constrained optimal control problems (OCPs) governed by the nonsmooth elliptic equations, in which the popular $$EQ_1^{rot}$$ E Q 1 r o t element is employed to approximate the state and adjoint state, and the piecewise constant element is used to approximate the control. Firstly, the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem. Secondly, the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate. Lastly, the methods are extended to some other well-known nonconforming elements.

中文翻译：

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非光滑椭圆方程约束最优控制问题的非协调有限元方法

在本文中，针对由非光滑椭圆方程控制的约束最优控制问题 (OCP) 提出了非一致性有限元方法 (FEM)，其中使用流行的 $$EQ_1^{rot}$$ EQ 1 rot 元素来近似状态和伴随状态，分段常数元素用于逼近控制。首先，通过引入一个辅助问题，得到了非光滑椭圆方程的收敛性和超收敛性。其次，通过建立负范数误差估计得到目标函数的面向目标的误差估计。最后，这些方法被扩展到其他一些众所周知的不合格元素。