In this paper, we consider the finite element approximation to a parabolic Dirichlet boundary control problem and establish new a priori error estimates. In the temporal semi-discretization we apply the DG(0) method for the state and the variational discretization for the control, and obtain the convergence rates $O(k^{\frac{1}{4}})$ and $O(k^{\frac{3}{4}-\varepsilon})$ $(\varepsilon>0)$ for the control for problems posed on polytopes with $y_0\in L^2(\Omega)$, $y_d\in L^2(I;L^2(\Omega))$ and smooth domains with $y_0\in H^{\frac{1}{2}}(\Omega)$, $y_d\in L^2(I;H^1(\Omega))\cap H^{\frac{1}{2}}(I;L^2(\Omega))$, respectively. In the fully discretization of the optimal control problem posed on polytopal domains, we apply the DG(0)-CG(1) method for the state and the variational discretization approach for the control, and derive the convergence order $O(k^{\frac{1}{4}} +h^{\frac{1}{2}})$, which improves the known results by removing the mesh size condition $k=O(h^2)$ between the space mesh size $h$ and the time step $k$. As a byproduct, we obtain a priori error estimate $O(h+k^{1\over 2})$ for the fully discretization of parabolic equations with inhomogeneous Dirichlet data posed on polytopes, which also improves the known error estimate by removing the above mesh size condition.

翻译：本文考虑抛物Dirichlet边界控制问题的有限元近似，并建立新的先验误差估计。在时间半离散化中，我们对状态采用DG(0)方法，对控制采用变分离散化，分别针对以下两种情况获得了控制的收敛速率$O(k^{\frac{1}{4}})$和$O(k^{\frac{3}{4}-\varepsilon})$ $(\varepsilon>0)$：当问题定义在多面体域上且$y_0\in L^2(\Omega)$、$y_d\in L^2(I;L^2(\Omega))$时，以及当问题定义在光滑域上且$y_0\in H^{\frac{1}{2}}(\Omega)$、$y_d\in L^2(I;H^1(\Omega))\cap H^{\frac{1}{2}}(I;L^2(\Omega))$时。针对定义在多面体域上的最优控制问题的全离散化，我们对状态采用DG(0)-CG(1)方法，对控制采用变分离散化方法，推导出收敛阶$O(k^{\frac{1}{4}} +h^{\frac{1}{2}})$，该结果通过消除空间网格尺寸$h$与时间步长$k$之间的网格尺寸条件$k=O(h^2)$改进了已知结果。作为副产品，我们获得了定义在多面体上具有非齐次Dirichlet数据的抛物方程全离散化的先验误差估计$O(h+k^{1\over 2})$，该结果同样通过消除上述网格尺寸条件改进了已知误差估计。