Stability and convergence analysis of the mixed finite element method for subdiffusive Oseen equations

This paper focuses on developing efficient numerical schemes for the subdiffusive Oseen equations with Caputo time-fractional derivative of order α∈(0,1). To address the computational challenges posed by the non-locality of the fractional derivative, we propose two distinct numerical methods, both employing mixed finite element methods for spatial discretization. Firstly, a variable-step fast L1 formula is introduced for time discretization, significantly reducing computational cost and memory requirements. The stability and convergence of this scheme are rigorously analyzed using a discrete Gronwall inequality, leading to optimal error estimates and verification of its energy stability. Secondly, a higher-accuracy uniform-step L1-2-3 formula is applied for time discretization, while the spatial direction is discretized via the mixed finite element method, with the stability and convergence of the scheme also demonstrated analytically. Finally, a series of numerical experiments are conducted to confirm the algorithms’ efficiency, convergence rates, and the validity of the theoretical findings, demonstrating their superior performance for solving fractional subdiffusion problems.