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 Grünwald-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.

在本文中，我们为带有非线性源项的分数瑞利-斯托克斯问题提出了一种改进的有限差分/有限元方法。使用二阶后向微分公式（BDF2）和加权平移的Grünwald-Letnikov差分（WSGD）公式分别离散一阶时间导数和时间分数阶导数。此外，提出了一种线性差分方案来近似非线性源项。结合空间方向的Galerkin有限元方法，我们提出了带有非线性源项的分数瑞利–斯托克斯问题的完全离散方案。基于一种新颖的分析技术，具有（（O（\ tau ^ {2} + h ^ {k + 1}）\）的\（L ^ {2} \）-范数的稳定性和收敛精度详细推导，并且该收敛顺序高于先前的工作。最后，通过一些数值例子验证了我们的理论结果。