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, global almost optimal error estimates in $L^2$- and $H^1$-norms are derived for the problem with initial data $u_0 \in H_0^1(\Omega)\cap H^2(\Omega)$. The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in $H^1$-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an $L^\infty$ error estimate is obtained for 2D problems that too with the initial condition is in $ H_0^1(\Omega)\cap H^2(\Omega)$. All results proved in this paper are valid uniformly as $\alpha\longrightarrow 1^{-}$, where $\alpha$ is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.

翻译：本文针对一类具有（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。所提方案基于时间方向分级网格上的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，推导了$L^2$和$H^1$范数下的全局几乎最优误差估计。本文方法的新颖之处在于通过管理分数阶导数的L1逼近与时间离散椭圆算子的相互作用，直接推导出$H^1$范数下的最优估计。此外，当椭圆算子为具有时变和空间变系数的自伴算子时，建立了超收敛结果，并由此在初始条件为$ H_0^1(\Omega)\cap H^2(\Omega)$的二维问题中获得了$L^\infty$误差估计。本文证明的所有结果在$\alpha\longrightarrow 1^{-}$时一致成立，其中$\alpha$为Caputo分数阶导数的阶数。数值实验验证了理论结果。