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AIMS Mathematics

2025, Volume 10, Issue 12: 29989-30016. doi: 10.3934/math.20251318
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Research article

Decoupled time filtered method for the time dependent Stokes–Biot problem

  • School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
  • Published: 22 December 2025

  • MSC : 76S05, 76D07, 65M6, 65M12

  • In this paper, we propose a novel decoupled time filtered finite element method for the coupled Stokes–Biot problem. The key innovation of our method is the combination of a decoupling strategy with a time filtered technique that can enhance computational efficiency while maintaining numerical accuracy. Specifically, applying the time filter to the backward Euler scheme elevates its temporal accuracy to second order. At each time step, we first solve the Stokes problem, then use the computed Stokes velocity to solve the Biot problem. We rigorously analyze the proposed scheme to establish its stability and derive error estimates. Furthermore, some numerical tests are presented to validate the theoretical findings and demonstrate the efficiency, accuracy, and robustness of our method.

    Citation: Liming Guo. Decoupled time filtered method for the time dependent Stokes–Biot problem[J]. AIMS Mathematics, 2025, 10(12): 29989-30016. doi: 10.3934/math.20251318

    Related Papers:

  • In this paper, we propose a novel decoupled time filtered finite element method for the coupled Stokes–Biot problem. The key innovation of our method is the combination of a decoupling strategy with a time filtered technique that can enhance computational efficiency while maintaining numerical accuracy. Specifically, applying the time filter to the backward Euler scheme elevates its temporal accuracy to second order. At each time step, we first solve the Stokes problem, then use the computed Stokes velocity to solve the Biot problem. We rigorously analyze the proposed scheme to establish its stability and derive error estimates. Furthermore, some numerical tests are presented to validate the theoretical findings and demonstrate the efficiency, accuracy, and robustness of our method.



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    [1] N. Jiang, H. Yang, SAV decoupled ensemble algorithms for fast computation of Stokes–Darcy flow ensembles, Comput. Methods Appl. Mech. Eng., 387 (2021), 114150. https://doi.org/10.1016/j.cma.2021.114150 doi: 10.1016/j.cma.2021.114150
    [2] M. Discacciati, A. Quarteroni, A. Valli, Robin–Robin domain decomposition methods for the Stokes–Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246–1268. https://doi.org/10.1137/06065091X doi: 10.1137/06065091X
    [3] W. Chen, M. Gunzburger, F. Hua, X. Wang, A parallel Robin–Robin domain decomposition method for the Stokes–Darcy system, SIAM J. Numer. Anal., 49 (2011), 1064–1084. https://doi.org/10.1137/080740556 doi: 10.1137/080740556
    [4] M. Mu, J. Xu, A two-grid method of a mixed Stokes–Darcy model for coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 45 (2007), 1801–1813. https://doi.org/10.1137/050637820 doi: 10.1137/050637820
    [5] M. Mu, X. Zhu, Decoupled schemes for a non-stationary mixed Stokes–Darcy model, Math. Comp., 79 (2010), 707–731. https://doi.org/10.1090/S0025-5718-09-02302-3 doi: 10.1090/S0025-5718-09-02302-3
    [6] W. Chen, M. Gunzburger, D. Sun, X. Wang, Efficient and long-time accurate second-order methods for the Stokes–Darcy system, SIAM J. Numer. Anal., 51 (2013), 2563–2584. https://doi.org/10.1137/120897705 doi: 10.1137/120897705
    [7] S. Badia, F. Nobile, C. Vergara, Robin–Robin preconditioned Krylov methods for fluid–structure interaction problems, Comput. Methods Appl. Mech. Eng., 198 (2009), 2768–2784. https://doi.org/10.1016/j.cma.2009.04.004 doi: 10.1016/j.cma.2009.04.004
    [8] S. Badia, A. Quaini, A. Quarteroni, Splitting methods based on algebraic factorization for fluid–structure interaction, SIAM J. Sci. Comput., 30 (2008), 1778–1805. https://doi.org/10.1137/070680497 doi: 10.1137/070680497
    [9] H. J. Bungartz, M. Schäfer, Fluid-structure interaction: modelling, simulation, optimisation, Lecture Notes in Computational Science and Engineering, Vol. 53, Springer Science & Business Media, 2006. https://doi.org/10.1007/3-540-34596-5
    [10] C. Michler, S. J. Hulshoff, E. H. Van Brummelen, R. de Borst, A monolithic approach to fluid–structure interaction, Comput. Fluids, 33 (2004), 839–848. https://doi.org/10.1016/j.compfluid.2003.06.006 doi: 10.1016/j.compfluid.2003.06.006
    [11] G. Guidoboni, R. Glowinski, N. Cavallini, S. Canic, Stable loosely-coupled-type algorithm for fluid–structure interaction in blood flow, J. Comput. Phys., 228 (2009), 6916–6937. https://doi.org/10.1016/j.jcp.2009.06.007 doi: 10.1016/j.jcp.2009.06.007
    [12] J. Xu, K. Yang, Well-posedness and robust preconditioners for discretized fluid–structure interaction systems, Comput. Methods Appl. Mech. Eng., 292 (2015), 69–91. https://doi.org/10.1016/j.cma.2014.09.034 doi: 10.1016/j.cma.2014.09.034
    [13] O. Oyekole, C. Trenchea, M. Bukač, A second-order in time approximation of fluid–structure interaction problem, SIAM J. Numer. Anal., 56 (2018), 590–613. https://doi.org/10.1137/17m1140054 doi: 10.1137/17m1140054
    [14] S. Badia, A. Quaini, A. Quarteroni, Coupling Biot and Navier–Stokes equations for modelling fluid–poroelastic media interaction, J. Comput. Phys., 228 (2009), 7986–8014. https://doi.org/10.1016/j.jcp.2009.07.019 doi: 10.1016/j.jcp.2009.07.019
    [15] A. Cesmelioglu, P. Chidyagwai, Numerical analysis of the coupling of free fluid with a poroelastic material, Numer. Methods Partial Differ. Equ., 36 (2020), 463–494. https://doi.org/10.1002/num.22437 doi: 10.1002/num.22437
    [16] J. Wen, Y. He, A strongly conservative finite element method for the coupled Stokes–Biot model, Comput. Math. Appl., 80 (2020), 1421–1442. https://doi.org/10.1016/j.camwa.2020.07.001 doi: 10.1016/j.camwa.2020.07.001
    [17] M. Zhou, R. Li, Z. Chen, A discontinuous Galerkin method for a coupled Stokes–Biot problem, J. Comput. Appl. Math., 451 (2024), 116086. https://doi.org/10.1016/j.cam.2024.116086 doi: 10.1016/j.cam.2024.116086
    [18] A. Cesmelioglu, J. J. Lee, S. Rhebergen, Hybridizable discontinuous Galerkin methods for the coupled Stokes–Biot problem, Comput. Math. Appl., 144 (2023), 12–33. https://doi.org/10.1016/j.camwa.2023.05.024 doi: 10.1016/j.camwa.2023.05.024
    [19] I. Ambartsumyan, E. Khattatov, I. Yotov, P. Zunino, A Lagrange multiplier method for a Stokes–Biot fluid–poroelastic structure interaction model, Numer. Math., 140 (2018), 513–553. https://doi.org/10.1007/s00211-018-0967-1 doi: 10.1007/s00211-018-0967-1
    [20] H. K. Wilfrid, A posteriori error analysis for a Lagrange multiplier method for a Stokes/Biot fluid-poroelastic structure interaction model, Abstr. Appl. Anal., 2021 (2021), 8877012. https://doi.org/10.1155/2021/8877012 doi: 10.1155/2021/8877012
    [21] O. Oyekole, M. Bukač, Second-order, loosely coupled methods for fluid-poroelastic material interaction, Numer. Methods Partial Differ. Equ., 36 (2020), 800–822. https://doi.org/10.1002/num.22452 doi: 10.1002/num.22452
    [22] L. Guo, W. Chen, A decoupled stabilized finite element method for the time–dependent Navier–Stokes/Biot problem, Math. Methods Appl. Sci., 45 (2022), 10749–10774. https://doi.org/10.1002/mma.8416 doi: 10.1002/mma.8416
    [23] L. Guo, W. Chen, Decoupled modified characteristic finite element method for the time-dependent Navier–Stokes/Biot problem, Numer. Methods Partial Differ. Equ., 38 (2022), 1684–1712. https://doi.org/10.1002/num.22830 doi: 10.1002/num.22830
    [24] H. Kunwar, H. Lee, K. Seelman, Second-order time discretization for a coupled quasi-Newtonian fluid-poroelastic system, Int. J. Numer. Methods Fluids, 92 (2020), 687–702. https://doi.org/10.1002/fld.4801 doi: 10.1002/fld.4801
    [25] C. Ager, B. Schott, M. Winter, W. A. Wall, A Nitsche-based cut finite element method for the coupling of incompressible fluid flow with poroelasticity, Comput. Methods Appl. Mech. Eng., 351 (2019), 253–280. https://doi.org/10.1016/j.cma.2019.03.015 doi: 10.1016/j.cma.2019.03.015
    [26] Z. Ge, J. Pang, J. Cao, Multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem, Numer. Methods Partial Differ. Equ., 39 (2023), 544–576. https://doi.org/10.1002/num.22903 doi: 10.1002/num.22903
    [27] S. Caucao, T. Li, I. Yotov, A multipoint stress-flux mixed finite element method for the Stokes–Biot model, Numer. Math., 152 (2022), 411–473. https://doi.org/10.1007/s00211-022-01310-2 doi: 10.1007/s00211-022-01310-2
    [28] R. Ruiz-Baier, M. Taffetani, H. D. Westermeyer, I. Yotov, The Biot–Stokes coupling using total pressure: formulation, analysis and application to interfacial flow in the eye, Comput. Methods Appl. Mech. Eng., 389 (2022), 114384. https://doi.org/10.1016/j.cma.2021.114384 doi: 10.1016/j.cma.2021.114384
    [29] X. Liu, Q. Song, Y. Li, J. Zhao, Three parameter-robust virtual element numerical schemes for a Stokes–Biot fluid-poroelastic structure interaction model, ESAIM: M2AN, 59 (2025), 2111–2139. https://doi.org/10.1051/m2an/2025055 doi: 10.1051/m2an/2025055
    [30] A. Guzel, W. Layton, Time filters increase accuracy of the fully implicit method, BIT Numer. Math., 58 (2018), 301–315. https://doi.org/10.1007/s10543-018-0695-z doi: 10.1007/s10543-018-0695-z
    [31] Y. Qin, Y. Hou, The time filter for the non-stationary coupled Stokes/Darcy model, Appl. Numer. Math., 146 (2019), 260–275. https://doi.org/10.1016/j.apnum.2019.07.015 doi: 10.1016/j.apnum.2019.07.015
    [32] Z. Tang, X. Jia, H. Feng, Decoupling time filter method for the non-stationary Navier-Stokes/Darcy model, Math. Methods Appl. Sci., 46 (2023), 3294–3305. https://doi.org/10.1002/mma.8691 doi: 10.1002/mma.8691
    [33] J. Wu, N. Li, X. Feng, Analysis of a filtered time-stepping finite element method for natural convection problems, SIAM J. Numer. Anal., 61 (2023), 837–871. https://doi.org/10.1137/21m1451476 doi: 10.1137/21m1451476
    [34] A. Cibik, F. G. Eroglu, S. Kaya, Analysis of second order time filtered backward Euler method for MHD equations, J. Sci. Comput., 82 (2020), 38. https://doi.org/10.1007/s10915-020-01142-y doi: 10.1007/s10915-020-01142-y
    [35] Y. Zhang, X. Yong, X. Du, Numerical analysis of time filter method for the stabilized incompressible diffusive Peterlin viscoelastic fluid model, Comput. Math. Appl., 168 (2024), 239–253. https://doi.org/10.1016/j.camwa.2024.07.002 doi: 10.1016/j.camwa.2024.07.002
    [36] W. Pei, The variable time-stepping DLN-ensemble algorithms for incompressible Navier-Stokes equations, Numer. Algor., 2025. https://doi.org/10.1007/s11075-025-02236-0 doi: 10.1007/s11075-025-02236-0
    [37] Y. Li, Y. Hou, W. Layton, H. Zhao, Adaptive partitioned methods for the time-accurate approximation of the evolutionary Stokes–Darcy system, Comput. Methods Appl. Mech. Eng., 364 (2020), 112923. https://doi.org/10.1016/j.cma.2020.112923 doi: 10.1016/j.cma.2020.112923
    [38] Y. Qin, Y. Wang, L. Chen, Y. Li, J. Li, A second-order adaptive time filter algorithm with different subdomain variable time steps for the evolutionary Stokes/Darcy model, Comput. Math. Appl., 150 (2023), 170–195. https://doi.org/10.1016/j.camwa.2023.09.027 doi: 10.1016/j.camwa.2023.09.027
    [39] J. Li, W. Song, Y. Qin, Z. Chen, An adaptive time filter algorithm with different subdomain time steps for super-hydrophobic proppants based on the 3d unsteady-state triple-porosity Stokes model, J. Sci. Comput., 101 (2024), 80. https://doi.org/10.1007/s10915-024-02716-w doi: 10.1007/s10915-024-02716-w
    [40] J. Li, Li. Chen, Q. Yi, Z. Chen A high-order, high-efficiency adaptive time filter algorithm for shale reservoir model based on coupled fluid flow with porous media flow, Commun. Nonlinear Sci. Numer. Simul., 146 (2025), 108771. https://doi.org/10.1016/j.cnsns.2025.108771 doi: 10.1016/j.cnsns.2025.108771
    [41] A. Cesmelioglu, Analysis of the coupled Navier–Stokes/Biot problem, J. Math. Anal. Appl., 456 (2017), 970–991. https://doi.org/10.1016/j.jmaa.2017.07.037 doi: 10.1016/j.jmaa.2017.07.037
    [42] M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155–164. https://doi.org/10.1063/1.1712886 doi: 10.1063/1.1712886
    [43] F. Brezzi, M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, Vol. 15, Springer, 1991. https://doi.org/10.1007/978-1-4612-3172-1
    [44] R. A. Adams, J. J. Fournier, Sobolev spaces, Elsevier, 2003.
    [45] P. G. Ciarlet, The finite element method for elliptic problems, Society for Industrial and Applied Mathematics, 2002.
    [46] V. DeCaria, W. Layton, H. Y. Zhao, A time-accurate, adaptive discretization for fluid flow problems, Int. J. Numer. Anal. Model., 17 (2020), 254–280.
    [47] J. G. Heywood, R. Rannacher, Finite-element approximation of the nonstationary Navier–Stokes problem. Part Ⅳ: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353–384. https://doi.org/10.1137/0727022 doi: 10.1137/0727022
    [48] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251–266. https://doi.org/10.1515/jnum-2012-0013 doi: 10.1515/jnum-2012-0013
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