Unconditionally optimal error estimates of two linearized Galerkin FEMs for the two-dimensional nonlinear fractional Rayleigh–Stokes problem

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 H²-norm. Secondly, we obtain the boundedness of the fully discrete finite element solution in L^∞-norm through the more detailed study of the error equation. Then, the optimal L²-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.