We consider an optimal control problem for the Poisson equation on a non-convex polygonal domain with the corner singularity. Previously, we proposed a novel algorithm for the accurate numerical solution for the Poisson equation on a polygonal domain with the domain singularity. Then, we investigated the error estimate and its efficient procedure for the numerical algorithm. In this article, we propose an efficient algorithm and perform an error estimate for a distributed optimal control problem of the Poisson equation. The solutions of the optimality system with such singularity have singular decompositions: regular part plus singular part for each state variable and adjoint variable. The coefficient of the singular function is usually called stress intensity factor and can be computed by the extraction formula. We introduced a modified optimality system which has “zero” stress intensity factors using this stress intensity factor, from whose solutions we can compute very accurate solution of the original optimality system simply by adding a singular part. We give a precise error analysis and provide numerical results which justify the results therein.

中文翻译：

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使用应力强度因子对具有拐角奇异性的最优控制问题的有限元方法进行误差估计

我们考虑具有角奇异性的非凸多边形域上的泊松方程的最优控制问题。此前，我们提出了一种新算法，用于求解具有奇异域的多边形域上的泊松方程的精确数值解。然后，我们研究了数值算法的误差估计及其有效过程。在本文中，我们提出了一种有效的算法，并对泊松方程的分布式最优控制问题进行了误差估计。具有这种奇异性的最优系统的解具有奇异分解：每个状态变量和伴随变量的正则部分加上奇异部分。奇异函数的系数通常称为应力强度因子，可以通过提取公式计算得到。我们引入了一个修正的最优系统，它使用这个应力强度因子具有“零”应力强度因子，我们可以从它的解中计算出原始最优系统的非常精确的解，只需添加一个奇异部分。我们给出了精确的误差分析并提供了数值结果来证明其中的结果是正确的。