This article is concerned with the nonconforming finite element method for distributed elliptic optimal control problems with pointwise constraints on the control and gradient of the state variable. We reduce the minimization problem into a pure state constraint minimization problem. In this case, the solution of the minimization problem can be characterized as fourth-order elliptic variational inequalities of the first kind. To discretize the control problem we have used the bubble enriched Morley finite element method. To ensure the existence of the solution to discrete problems three bubble functions corresponding to the mean of the edge are added to the discrete space. We derive the error in the state variable in $H^2$-type energy norm. Numerical results are presented to illustrate our analytical findings.
翻译：本条涉及分布式椭圆最佳控制问题不兼容的限定要素方法,对状态变量的控制和梯度有点限制。我们将最小化问题减为纯状态限制最小化问题。在这种情况下,最小化问题的解决可被定性为第一种第一种水平的椭圆变异性不平等。为了将控制问题分解,我们使用了泡沫浓缩莫利限制要素方法。为了确保对离散问题的解决办法的存在,在离散空间中增加了与边缘平均值相对应的三个泡沫功能。我们从状态变量中得出以$H$2$为型的能源规范的错误。提供了数字结果,以说明我们的分析结论。
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