We derive optimal order a posteriori error estimates in the $L^\infty(L^2)$ and $L^1(L^2)$-norms for the fully discrete approximations of time fractional parabolic differential equations. For the discretization in time, we use the $L1$ methods, while for the spatial discretization, we use standard conforming finite element methods. The linear and quadratic space-time reconstructions are introduced, which are generalizations of the elliptic space reconstruction. Then the related a posteriori error estimates for the linear and quadratic space-time reconstructions play key roles in deriving global and pointwise final error estimates. Numerical experiments verify and complement our theoretical results.

翻译：我们针对时间分数阶抛物型微分方程的全离散逼近，推导了$L^\infty(L^2)$和$L^1(L^2)$范数下的最优阶后验误差估计。时间离散采用$L1$方法，空间离散采用标准协调有限元方法。引入线性与二次时空重构方法，这些方法是椭圆空间重构的推广。在此基础上，线性与二次时空重构的相关后验误差估计在全局和逐点最终误差估计的推导中起关键作用。数值实验验证并补充了我们的理论结果。