An improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term

Abstract

In this paper, we propose an improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term. The second-order backward differentiation formula (BDF2) and weighted and shifted Grunwald-Letnikov difference (WSGD) formula are employed to discretize first-order time derivative and the time fractional-order derivative, respectively. Moreover, a linearized difference scheme is proposed to approximate the nonlinear source term. Together with the Galerkin finite element method in the space direction, we present a fully discrete scheme for the fractional Rayleigh–Stokes problem with a nonlinear source term. Based on a novel analytical technique, the stability and the convergence accuracy in $$L^{2}$$ -norm with $$O(\tau ^{2}+h^{k+1})$$ are derived in detail, and this convergence order is higher than the previous work. Finally, some numerical examples are presented to validate our theoretical results.