In this paper, an anisotropic bilinear finite element method is constructed for the elliptic boundary layer optimal control problems. Supercloseness properties of the numerical state and numerical adjoint state in a $ \epsilon $-norm are established on anisotropic meshes. Moreover, an interpolation type post-processed solution is shown to be superconvergent of order $ O(N^{-2}) $, where the total number of nodes is of $ O(N^2) $. Finally, numerical results are provided to verify the theoretical analysis.

中文翻译：

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椭圆边界层最优控制问题的非均匀各向异性有限元

针对椭圆边界层的最优控制问题，构造了各向异性双线性有限元方法。在各向异性网格上建立$ \ε范数中的数值状态和数值伴随状态的超闭合性。此外，内插类型的后处理解决方案显示为阶$ O（N ^ {-2}）$的超收敛，其中节点总数为$ O（N ^ 2）$。最后，提供数值结果以验证理论分析。