This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy regularization. The regularity of the solution is carefully carved out exploiting weighted Sobolev and Hölder spaces. This allows to derive a sharp relation between the convergence rates for the approximation and the structure of the geometry, more precisely, the largest opening angle at the vertices of polygonal domains. Numerical experiments confirm that the derived convergence rates are sharp.

中文翻译：

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具有能量正则化的变分正态导数和狄利克雷控制问题的误差估计

本文处理变分法向导数的有限元逼近的误差估计，因此，涉及能量正则化的狄利克雷边界控制问题的有限元逼近的误差估计。利用加权的 Sobolev 和 Hölder 空间仔细地雕刻出解决方案的规律性。这允许推导出近似收敛速度与几何结构之间的密切关系，更准确地说，是多边形域顶点处的最大张角。数值实验证实，导出的收敛速度非常快。