Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Gr\"{o}nwall inequality, optimal error estimates in $L^2$- and $H^1$-norms are derived for the problem with initial data $u_0 \in H_0^1(\Omega)\cap H^2(\Omega)$. Under higher regularity condition $u_0 \in \dot{H}^3(\Omega)$, a super convergence result is established and as a consequence, $L^\infty$ error estimate is obtained for 2D problems. Numerical experiments are presented to validate our theoretical findings.

翻译：针对一类具有(时空)变系数非自伴椭圆部分的时间分数阶线性偏微分/积分微分方程，研究了非均匀隐式-显式L1有限元方法(IMEX-L1-FEM)的稳定性与最优收敛性分析。所提出的方案基于时间方向渐变网格上的IMEX-L1方法与空间方向有限元方法的结合。借助离散分数阶Grönwall不等式，针对初始数据$u_0 \in H_0^1(\Omega)\cap H^2(\Omega)$的问题，导出了$L^2$和$H^1$范数下的最优误差估计。在更高正则性条件$u_0 \in \dot{H}^3(\Omega)$下，建立了超收敛结果，并由此获得了二维问题的$L^\infty$误差估计。数值实验验证了我们的理论发现。