  This article presents a finite element scheme with Newton's method for
solving the time-fractional nonlinear diffusion equation. For time
discretization, we use the fractional Crank-Nicolson scheme based on backward
Euler convolution quadrature. We discuss the existence-uniqueness results for
the fully-discrete problem. Discrete fractional Gronwall type inequality for
the backward Euler convolution quadrature which is used to approximate the
Riemann-Liouville fractional derivative is established by using the idea of
D.Li et al. given in [11]. A priori error estimate for the fully-discrete
problem in $L^2(\Omega)$ norm is derived. Numerical results based on finite
element scheme are provided to validate the theoretical estimates.
