In this paper, two linearized Galerkin finite element methods, which are based on the L1 approximation and the WSGD operator, respectively, are proposed to solve the nonlinear fractional Rayleigh-Stokes problem. In order to obtain the unconditionally optimal error estimate, we firstly introduce a time-discrete elliptic equation, and derive the unconditional error estimate between the exact solution and the solution of the time-discrete system in $H^2$-norm. Secondly, we obtain the boundedness of the fully discrete finite element solution in $L^{\infty }$-norm through the more detailed study of the error equation. Then, the optimal $L^2$-norm error estimate is derived for the fully discrete system without any restriction conditions on the time step size. Finally, some numerical experiments are presented to confirm the theoretical results, showing that the two linearized schemes given in this paper are efficient and reliable.

本文提出了分别基于L 1逼近和WSGD算子的两种线性化Galerkin有限元方法来解决非线性分数瑞利-斯托克斯问题。为了获得无条件的最优误差估计，我们首先引入了一个时离散椭圆方程，并在时滞系统的精确解和解之间推导了无条件误差估计。$H^{\mathrm{2个}}$-规范。其次，我们得到了全离散有限元解的有界性${\mathrm{大号}}^{\infty }$-通过对误差方程的更详细研究来规范。然后，最优${\mathrm{大号}}^{\mathrm{2个}}$-标准误差估计是针对完全离散的系统得出的，而时间步长没有任何限制条件。最后，通过一些数值实验验证了理论结果，表明本文给出的两种线性化方案是有效且可靠的。