Fractional Crank-Nicolson-Galerkin Finite Element Scheme for the Time-Fractional Nonlinear Diffusion Equation

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.