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All issues Volume 59 / No 4 (July-August 2025) ESAIM: M2AN, 59 4 (2025) 2111-2139 References
Open Access
Issue
ESAIM: M2AN
Volume 59, Number 4, July-August 2025
Page(s) 2111 - 2139
DOI https://doi.org/10.1051/m2an/2025055
Published online 23 July 2025
  1. I. Ambartsumyan, E. Khattatov, I. Yotov and P. Zunino, A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model. Numer. Math. 140 (2018) 513–553. [Google Scholar]
  2. I. Ambartsumyan, V.J. Ervin, T. Nguyen and I. Yotov, A nonlinear Stokes-Biot model for the interaction of a non-Newtonian fluid with poroelastic media. ESAIM Math. Model. Numer. Anal. 53 (2019) 1915–1955. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  3. P.F. Antonietti, L. Beirão da Veiga, D. Mora and M. Verani, A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52 (2014) 386–404. [Google Scholar]
  4. D.N. Arnold, J. Douglas and C.P. Gupta, A family of high order mixed fine element methods for plane elasticity. Numer. Math. 45 (1984) 1–22. [Google Scholar]
  5. B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method. ESAIM Math. Model. Numer. Anal. 50 (2016) 879–904. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  6. S. Badia, A. Quaini and A. Quarteroni, Coupling biot and Navier–Stokes equations for modelling fluid-poroelastic media interaction. J. Comput. Phys. 228 (2009) 7986–8014. [Google Scholar]
  7. Y. Bazilevs, K. Takizawa and T.E. Tezduyar, Computational Fluid-Structure Interaction: Methods and Applicattions. John Wiley & Sons, New York (2013). [Google Scholar]
  8. L. Beirão da Veiga and G. Manzini, A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34 (2014) 759–781. [Google Scholar]
  9. L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods. Math. Mod. Meth. Appl. Sci. 23 (2013) 199–214. [CrossRef] [Google Scholar]
  10. L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, The Hitchhiker’s guide to the virtual element method. Math. Mod. Meth. Appl. Sci. 24 (2014) 1541–1573. [Google Scholar]
  11. L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, H(div) and H(curl)-conforming virtual element methods. Numer. Math. 133 (2016) 303–332. [CrossRef] [MathSciNet] [Google Scholar]
  12. L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes. Math. Mod. Meth. Appl. Sci. 26 (2016) 729–750. [Google Scholar]
  13. L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM Math. Model. Numer. Anal. 50 (2016) 727–747. [Google Scholar]
  14. L. Beirão da Veiga, F. Dassi and A. Russo, High-order virtual element method on polyhedral meshes. Comput. Math. Appl. 74 (2017) 1110–1122. [Google Scholar]
  15. L. Beirão da Veiga, C. Lovadina and G. Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model. Numer. Anal. 51 (2017) 509–535. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  16. L. Beirão da Veiga, D. Mora and G. Vacca, The Stokes complex for virtual elements with application to Navier–Stokes flows. J. Sci. Comput. 81 (2019) 990–1018. [Google Scholar]
  17. E.A. Bergkamp, C.V. Verhoosel, J.J.C. Remmers and D.M.J. Smeulders, A staggered finite element procedure for the coupled Stokes-Biot system with fluid entry resistance. Comput. Geosci. 24 (2020) 1497–1522. [CrossRef] [MathSciNet] [Google Scholar]
  18. W.M. Boon, M. Hornkj∅l, M. Kuchta, K.-A. Mardal and R. Ruiz-Baier, Parameter-robust methods for the Biot-Stokes interfacial coupling without Lagrange multipliers. J. Comput. Phys. 467 (2022) 111464. [Google Scholar]
  19. K.E. Brenan, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia (1995). [Google Scholar]
  20. F. Brezzi, R.S. Falk and L.D. Marini, Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. 48 (2014) 1227–1240. [Google Scholar]
  21. M. Bukač, A loosely-coupled scheme for the interaction between a fluid, elastic structure and poroelastic material. J. Comput. Phys. 313 (2016) 377–399. [Google Scholar]
  22. M. Bukač, I. Yotov, R. Zakerzadeh and P. Zunino, Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach. Comput. Methods Appl. Mech. Eng. 292 (2015) 138–170. [CrossRef] [Google Scholar]
  23. R. Burger, S. Kumar, D. Mora, R. Ruiz-Baier and N. Verma, Virtual element methods for the three-field formulation of time-dependent linear poroelasticity. Adv. Comput. Math. 47 (2021) 2. [Google Scholar]
  24. Y. Cao, M. Gunzburger, X. He and X. Wang, Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes-Darcy systems. Math. Comput. 83 (2014) 1617–1644. [Google Scholar]
  25. S. Caucao, T. Li and I. Yotov, A cell-centered finite volume method for the Navier–Stokes/Biot Model, in Finite Volumes for Complex Applications IX – Methods, Theoretical Aspects, Examples, FVCA 2020, edited by R. Klöfkorn, E. Keilegavlen, F.A. Radu and J. Fuhrmann. Vol. 323 of Springer Proceedings in Mathematics & Statistics. Springer, Cham (2020). [Google Scholar]
  26. S. Caucao, T. Li and I. Yotov, A multipoint stress-flux mixed finite element method for the Stokes-Biot model Numer. Math. 152 (2022) 411–473. [Google Scholar]
  27. A. Cesmelioglu, Analysis of the coupled Navier–Stokes/Biot problem. J. Math. Anal. Appl. 456 (2017) 970–991. [Google Scholar]
  28. A. Cesmelioglu and P. Chidyagwai, Numerical analysis of the coupling of free fluid with a poroelastic material. Numer. Methods Part. Differ. Equ. 36 (2020) 463–494. [CrossRef] [Google Scholar]
  29. A. Cesmelioglu, J.J. Lee and S. Rhebergen, Hybridizable discontinuous Galerkin methods for the coupled Stokes-Biot problem. Comput. Math. Appl. 144 (2023) 12–33. [Google Scholar]
  30. L. Chen and F. Wang, A divergence free weak virtual element method for the Stokes problem on polytopal meshes. J. Sci. Comput. 78 (2019) 864–886. [Google Scholar]
  31. P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis. Vol. II. Elsevier, North-Holland, Amsterdam (1991) 17–351. [Google Scholar]
  32. J. Coulet, I. Faille, V. Girault, N. Guy and F. Nataf, A fully coupled scheme using virtual element method and finite volume for poroelasticity. Comput. Geosci. 24 (2020) 381–403. [Google Scholar]
  33. M. Discacciati, E. Miglio and A. Quarteroni, Mathematical and numerical models for coupling surface and ground-water flows. Appl. Numer. Math. 43 (2002) 57–74. [Google Scholar]
  34. L. Formaggia, A. Quarteroni and A. Veneziani, Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System. Vol. 1. Springer, Milan (2010). [Google Scholar]
  35. G.N. Gatica, S. Meddahi and R. Oyarzúa, A conforming mixed finite-element method for the coupling of fluid flow with porous media flow. IMA J. Numer. Anal. 29 (2009) 86–108. [Google Scholar]
  36. V. Girault and B. Rivière, DG approximation of coupled Navier–Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Numer. Anal. 47 (2009) 2052–2089. [Google Scholar]
  37. V. Girault, M.F. Wheeler, B. Ganis and M.E. Mear, A lubrication fracture model in a poroelastic medium. Math. Models Methods Appl. Sci. 25 (2015) 587–645. [Google Scholar]
  38. L.M. Guo and W.B. Chen, A decoupled stabilized finite element method for the time-dependent Navier–Stokes/Biot problem. Math. Meth. Appl. Sci. 45 (2022) 10749–10774. [Google Scholar]
  39. L.M. Guo and W.B. Chen, Decoupled modified characteristic finite element method for the time-dependent Navier–Stokes/Biot problem. Numer. Methods Part. Differ. Equ. 38 (2022) 1684–1712. [Google Scholar]
  40. J. Guo and M. Feng, A new projection-based stabilized virtual element method for the Stokes problem. J. Sci. Comput. 85 (2020) 16. [Google Scholar]
  41. J. Guo and M.F. Feng, A robust and mass conservative virtual element method for linear three-field poroelasticity. J. Sci. Comput. 92 (2022) 95. [Google Scholar]
  42. G. Hou, J. Wang and A. Layton, Numerical methods for fluid-structure interaction-a review. Commun. Comput. Phys. 12 (2012) 337–377. [Google Scholar]
  43. H. Kunwar, H. Lee and K. Seelman, Second-order time discretization for a coupled quasi-Newtonian fluid-poroelastic system. Int. J. Numer. Methods Fluids 92 (2019) 687–702. [Google Scholar]
  44. W.J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2003) 2195–2218. [Google Scholar]
  45. T. Li and I. Yotov, A mixed elasticity formulation for fluid-poroelastic structure interaction. ESAIM Math. Model. Numer. Anal. 56 (2022) 1–40. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  46. T. Li, S. Caucao and I. Yotov, An augmented fully mixed formulation for the quasistatic Navier–Stokes-Biot model. IMA J. Numer. Anal. 44 (2024) 1153–1210. [Google Scholar]
  47. X. Liu and Z.X. Chen, A virtual element method for overcoming locking phenomena in Biot’s consolidation model. ESAIM Math. Model. Numer. Anal. 57 (2023) 3007–3027. [Google Scholar]
  48. X. Liu, R. Li and Z.X. Chen, A virtual element method for the coupled Stokes-Darcy problem with the Beaver-Joseph-Saffman interface condition. Calcolo 56 (2019) 48. [Google Scholar]
  49. X. Liu, R. Li and Y.F. Nie, A divergence-free reconstruction of the nonconforming virtual element method for the Stokes problem. Comput. Methods Appl. Mech. Eng. 372 (2020) 113351. [CrossRef] [Google Scholar]
  50. A. Lozovskiy, M.A. Olshanskii and Y.V. Vassilevski, A finite element scheme for the numerical solution of the Navier–Stokes/Biot coupled problem. Rus. J. Numer. Anal. Math. Model. 37 (2022) 159–174. [Google Scholar]
  51. L. Mascotto, Ill-conditioning in the virtual element method: stabilizations and bases. Numer. Methods Part. Differ. Equ. 34 (2018) 1258–1281. [Google Scholar]
  52. C. Michler, S. Hulshoff, E. Van Brummelen and R. De Borst, A monolithic approach to fluid-structure interaction. Comput. Fluids 33 (2004) 839–848. [Google Scholar]
  53. M. Mu and X. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model. Math. Comput. 79 (2010) 707–731. [Google Scholar]
  54. T.Q. Nguyen, Modelling of flow and transport of non-Newtonian fluids interacting with poroelastic media. Doctoral dissertation, University of Pittsburgh (2019). [Google Scholar]
  55. O. Oyekole and M. Bukač, Second-order, loosely coupled methods for fluid-poroelastic material interaction. Numer. Methods Part. Differ. Equ. 36 (2020) 800–822. [Google Scholar]
  56. A. Quarteroni, A. Veneziani and P. Zunino, Mathematical and numerical modeling of solute dynamics in blood flow and arterial walls. SIAM J. Numer. Anal. 39 (2001) 1488–1511. [Google Scholar]
  57. T. Richter, Fluid-Structure Interactions: Models, Analysis and Finite Elements. Vol. 118. Springer, Cham (2017). [Google Scholar]
  58. B. Rivière and I. Yotov, Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal. 42 (2005) 1959–1977. [Google Scholar]
  59. R. Ruiz-Baier, M. Taffetani, H.D. Westermeyer and 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. [CrossRef] [Google Scholar]
  60. R.E. Showalter, Poroelastic filtration coupled to Stokes flow, in Control Theory of Partial Differential Equations. Vol. 242 of Lecture Notes in Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton (2005) 229–241. [Google Scholar]
  61. X.L. Tang, Z.B. Liu, B.J. Zhang and M.F. Feng, On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity. ESAIM Math. Model. Numer. Anal. 55 (2021) S909–S939. [Google Scholar]
  62. H.F. Wang, Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press, NewJersey (2017). [Google Scholar]
  63. X. Wang and H. Rui, The locking-free finite difference method based on staggered grids for the coupled Stokes-Biot problem. Int. J. Comput. Math. 99 (2022) 2042–2068. [Google Scholar]
  64. G. Wang, F. Wang, L. Chen and Y.N. He, A divergence free weak virtual element method for the Stokes-Darcy problem on general meshes. J. Sci. Comput. 344 (2019) 998–1020. [Google Scholar]
  65. F. Wang, M.C. Cai, G. Wang and Y.P. Zeng, A mixed virtual element method for Biot’s consolidation model. Comput. Math. Appl. 126 (2022) 31–42. [Google Scholar]
  66. H.Y. Wei, X.H. Huang and A. Li, Piecewise divergence-free nonconforming virtual elements for Stokes problem in any dimensions. SIAM J. Numer. Anal. 59 (2021) 1835–1856. [Google Scholar]
  67. J. Wen and Y. He, A strongly conservative finite element method for the coupled Stokes-Biot model. Comput. Math. Appl. 80 (2020) 1421–1442. [CrossRef] [MathSciNet] [Google Scholar]
  68. J. Wen, J. Su, Y. He and H. Chen, Discontinuous Galerkin method for the coupled Stokes-Biot model. Numer. Methods Part. Differ. Equ. 37 (2021) 383–405. [Google Scholar]
  69. H.K. Wilfrid, Nonconforming finite element methods for a Stokes/Biot fluid-poroelastic structure interaction model. Results Appl. Math. 7 (2020) 100127. [Google Scholar]
  70. J. Xu and K. Yang, Well-posedness and robust preconditioners for discretized fluid-structure interaction systems. Comput. Methods Appl. Mech. Eng. 292 (2015) 69–91. [Google Scholar]
  71. R. Zakerzadeh, M. Bukac and P. Zunino, Computational analysis of energy distribution of coupled blood flow and arterial deformation. Int. J. Adv. Eng. Sci. Appl. Math. 8 (2016) 70–85. [Google Scholar]
  72. J.K. Zhao, B. Zhang, S.P. Mao and S.C. Chen, The divergence-free nonconforming virtual element for the Stokes problem. SIAM J. Numer. Anal. 57 (2019) 2730–2759. [CrossRef] [MathSciNet] [Google Scholar]

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