We study the space-time finite element discretizations for time fractional parabolic integro-differential equations in a bounded convex polygonal domain in \({\mathbb {R}}^d (d=1,2,3)\). Both spatially semidiscrete and fully discrete finite element approximations are considered and analyzed. We use piecewise linear and continuous finite elements to approximate the space variable whereas the time discretization uses two fully discrete schemes based on the convolution quadrature, namely the backward Euler and the second-order backward difference. For the spatially discrete scheme, optimal order a priori error estimates are derived for smooth initial data, i.e., when \(u_0\in H_0^1(\varOmega )\cap H^2(\varOmega )\). Moreover, for the homogeneous problem, almost optimal error estimates for positive time are established for nonsmooth initial data, i.e., when the initial function \(u_0\) is only in \( L^2(\varOmega )\). The error estimates for the fully discrete methods are shown to be optimal in time for both smooth and nonsmooth initial data under the specific choice of the kernel operator in the integral. Finally, we provide some numerical illustrations to verify our theoretical analysis.

我们研究\（{\ mathbb {R}} ^ d（d = 1,2,3）\）中有界凸多边形域中时间分数抛物型积分微分方程的时空有限元离散化。在空间上半离散和完全离散的有限元近似都被考虑和分析。我们使用分段线性和连续有限元来近似空间变量，而时间离散化使用基于卷积正交的两个完全离散的方案，即后向欧拉和二阶后向差分。对于空间离散方案，对于平滑的初始数据，即，当\（u_0 \ H_0 ^ 1（\ varOmega）\ cap H ^ 2（\ varOmega）\）时，得出最优阶先验误差估计。。此外，对于齐次问题，对于不平滑的初始数据，即，当初始函数\（u_0 \）仅在\（L ^ 2（\ varOmega）\中时，几乎可以确定正时间的最佳误差估计。对于在积分中内核运算符的特定选择下，对于平滑和非平滑初始数据而言，完全离散方法的误差估计在时间上均显示为最佳。最后，我们提供一些数值例证来验证我们的理论分析。