  In this paper, we develop a linearized fractional Crank-Nicolson-Galerkin FEM
for Kirchhoff type quasilinear time-fractional integro-differential equation
$\left(\mathcal{D}^{\alpha}\right)$. In general, the solutions to the
time-fractional problems exhibit a weak singularity at time $t=0$. This
singular behavior of the solutions is taken into account while deriving the
convergence estimates of the developed numerical scheme. We prove that the
proposed numerical scheme has an accuracy rate of $O(M^{-1}+N^{-2})$ in
$L^{\infty}(0,T;L^{2}(\Omega))$ as well as in
$L^{\infty}(0,T;H^{1}_{0}(\Omega))$, where $M$ and $N$ are the degrees of
freedom in the space and time directions respectively. A numerical experiment
is presented to verify the theoretical results.
