In this paper, we present a two‐grid mixed finite element scheme for distributed optimal control governed by general elliptic equations. –P₁ mixed finite elements are used for the discretization of the state and co‐state variables, whereas piecewise constant function is used to approximate the control variable. We first use a new approach to obtain the superclose property between the centroid interpolation and the numerical solution of the optimal control u with order h² under the low regularity. Based on the superclose property, we derive the optimal a priori error estimates. Then, using a postprocessing projection operator, we get a second‐order superconvergent result for the control u. Next, we construct a two‐grid mixed finite element scheme and analyze a priori error estimates. In the two‐grid scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, a numerical example is presented to verify the theoretical results.

中文翻译：

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低规则性的一般椭圆最优控制问题的–P1混合有限元逼近的两网格方法

在本文中，我们提出了一种由通用椭圆方程控制的两网格混合有限元方案，用于分布式最优控制。– P ₁混合有限元用于状态变量和共状态变量的离散化，而分段常数函数用于近似控制变量。我们首先使用一种新方法来获得质心插值和低规则性下h ²阶最优控制u的数值解之间的超闭合特性。基于superclose属性，我们得出最优的先验误差估计。然后，使用后处理投影算子，我们得到控制u的二阶超收敛结果。。接下来，我们构造一个两网格混合有限元方案并分析先验误差估计。在双网格方案中，将细网格上的椭圆最优控制问题的解简化为一个粗得多的网格上的椭圆最优控制问题的解，以及细网格上的线性代数系统的解及其结果解决方案仍保持渐近最佳精度。最后，通过数值例子验证了理论结果。