Error Estimates for a Linearized Fractional Crank-Nicolson FEM for Kirchhoff type Quasilinear Subdiffusion Equation with Memory

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 $L^{\mathrm{\infty }}(0,T;L^2(\mathrm{\Omega }))$ as well as in $L^{\mathrm{\infty }}(0,T;H_0^1(\mathrm{\Omega }))$, 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.