  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.
