Galerkin finite element schemes with fractional Crank–Nicolson method for the coupled time-fractional nonlinear diffusion system

Abstract

This paper deals with two fractional Crank–Nicolson–Galerkin finite element schemes for coupled time-fractional nonlinear diffusion system. The first scheme is iterative and is based on Newton’s method, while the other one is a linearized scheme. Existence-uniqueness results of the fully discrete solution for both schemes are discussed. In addition, a priori bounds and a priori error estimates are derived for proposed schemes using a new discrete fractional Gronwall-type inequality. Both the schemes yield $$O({\varDelta } t^2)$$ accuracy in time and hence, superior to $$O({\varDelta } t^{2-\alpha })$$ accurate L1 scheme existing in the literature. Moreover, three different numerical examples are provided to illustrate the theoretical estimates .