标题： 一种抛物线狄利克雷边界控制问题的新误差分析 显示英文标题

标题： A new error analysis for parabolic Dirichlet boundary control problems

摘要： 在本文中，我们考虑抛物线狄利克雷边界控制问题的有限元逼近，并建立新的先验误差估计。在时间半离散化中，我们对状态应用DG(0)方法，对控制应用变分离散化，并得到在多面体上具有$y_0\in L^2(\Omega)$，$y_d\in L^2(I;L^2(\Omega))$和光滑域上具有$y_0\in H^{\frac{1}{2}}(\Omega)$，$y_d\in L^2(I;H^1(\Omega))\cap H^{\frac{1}{2}}(I;L^2(\Omega))$的控制的收敛率$O(k^{\frac{1}{4}})$和$O(k^{\frac{3}{4}-\varepsilon})$$(\varepsilon>0)$。 在针对多面体域上的最优控制问题的完全离散化中，我们对状态应用DG(0)-CG(1)方法，对控制采用变分离散化方法，并推导出收敛阶$O(k^{\frac{1}{4}} +h^{\frac{1}{2}})$，这通过去除空间网格大小$h$与时间步长$k$之间的网格尺寸条件$k=O(h^2)$改进了已知结果。 作为副产品，我们得到了针对多面体上具有非齐次狄利克雷数据的抛物方程完全离散化的先验误差估计$O(h+k^{1\over 2})$，这也通过去除上述网格尺寸条件改进了已知的误差估计。 显示英文摘要

摘要： 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.