Finite Element Approximations of an Optimal Control Problem with Integral State Constraint
Finite Element Approximations of an Optimal Control Problem with Integral State Constraint
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DOI：
10.1137/080737095
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发表时间：
2010-07
期刊：
影响因子：
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通讯作者：
Wenbin Liu;Danping Yang;Lei Yuan;Chaoqun Ma
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文献类型：
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作者：
Wenbin Liu;Danping Yang;Lei Yuan;Chaoqun Ma
考虑一类椭圆型偏微分方程解的积分状态约束最优控制问题及其有限元逼近。有限元逼近是在多重网格上构造的。得到了有限元逼近的$L^2$-范数的先验误差估计。在此基础上，证明了一些超收敛结果。在这些超收敛结果的基础上，给出了几乎最优的L范数误差估计。然后提出了一些恢复算法来产生梯度型的后验误差估计器。针对有限元问题，提出了一种简单高效的迭代梯度投影算法，并证明了该算法的收敛速度。通过数值算例验证了理论分析的正确性。
An integral state-constrained optimal control problem governed by an elliptic partial differential equation and its finite element approximation are considered. The finite element approximation is constructed on multimeshes. An $L^2$-norm a priori error estimate of the finite element approximation is obtained. Further, some superconvergence results are proved. Based on these superconvergence results, almost optimal $L^\infty$-norm error estimates are derived. Some recovery algorithms are then proposed to produce a posteriori error estimators of gradient type. To solve the finite element system, a simple and yet efficient iterative gradient projection algorithm is proposed and its convergence rate is proved. Some numerical examples are performed to confirm theoretical analysis.