Our aim is to study a posteriori error estimates for the finite element method of the parabolic boundary control problems on a bounded convex polygonal domain. For discretization, piecewise linear and continuous finite elements are used to approximate the state and the adjoint-state variables, while piecewise constant functions are employed to approximate the control variable. The backward Euler implicit scheme is applied to discretize the time derivative. An adaptation of a novel elliptic reconstruction technique plays a key role in deriving the error estimates. The residual type a posteriori error estimates for the state and adjoint-state variables are derived in the $L^{\infty }(0,T;H^1(\mathrm{\Omega }))$-norm. Furthermore, an error bound for the control variable is established in the $L^{\infty }(0,T;L^2(\mathrm{\Gamma }))$-norm. The numerical experiments illustrate the performance of the derived estimators. The adaptive mesh generated via the error indicators is very less in comparison to the uniform mesh, which shows the effectiveness of our derived estimators.

我们的目的是研究有界凸多边形域上抛物线边界控制问题的有限元方法的后验误差估计。对于离散化，分段线性和连续有限元用于近似状态和伴随状态变量，而分段常数函数用于近似控制变量。应用后向欧拉隐式格式来离散化时间导数。一种新颖的椭圆重建技术的适应在推导误差估计方面发挥着关键作用。状态和伴随状态变量的残差类型后验误差估计是在$L^{无穷大}（0,\mathrm{时间};H^1（\mathrm{\Omega }））$-规范。此外，控制变量的误差界限在$L^{无穷大}（0,\mathrm{时间};L^2（\mathrm{\gamma }））$-规范。数值实验说明了推导估计器的性能。与均匀网格相比，通过误差指示器生成的自适应网格要少得多，这显示了我们导出的估计器的有效性。