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

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  • Wang, Zhen
  • Wang, Yujie
  • Zhu, Rui

Abstract

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.

Suggested Citation

  • Wang, Zhen & Wang, Yujie & Zhu, Rui, 2026. "Stability and convergence analysis of the mixed finite element method for subdiffusive Oseen equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 245(C), pages 483-511.
    • Handle: RePEc:eee:matcom:v:245:y:2026:i:c:p:483-511
    DOI: 10.1016/j.matcom.2026.01.024
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    References listed on IDEAS

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    1. Zhao, Jingjun & Zhao, Wenjiao & Xu, Yang, 2021. "Lagrange nodal discontinuous Galerkin method for fractional Navier-Stokes equations," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    2. Zhang, Jun & Wang, JinRong, 2018. "Numerical analysis for Navier–Stokes equations with time fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 481-489.
    3. Lu, Zuliang & Zhang, Shuhua, 2017. "L∞-error estimates of rectangular mixed finite element methods for bilinear optimal control problem," Applied Mathematics and Computation, Elsevier, vol. 300(C), pages 79-94.
    4. Zhang, Tong & Li, ShiShun, 2017. "A posteriori error estimates of finite element method for the time-dependent Navier–Stokes equations," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 13-26.
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    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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