Skip to main content
SpringerLink
Log in

A Robust and Mass Conservative Virtual Element Method for Linear Three-field Poroelasticity

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We present and analyze a robust and mass conservative virtual element method for the three-field formulation of Biot’s consolidation problem in poroelasticity. The displacement and fluid flux are respectively approximated by enriched \({\varvec{H}}({\text {div}})\) virtual elements and \({\varvec{H}}({\text {div}})\) virtual elements, while the pressure is discretized by piecewise polynomial functions. Optimal a priori error estimates are obtained, including the semi-discrete scheme and the fully-discrete scheme with the implicit Euler approximation in time. Moreover, our method achieves robustness with respect to the constants hidden in the error estimates, even for the Lamé coefficient tending to infinity and the arbitrarily small constrained specific storage coefficient, and therefore it is free of both volumetric (Poisson) locking and nonphysical pressure oscillations. Meanwhile, it also conserves pointwise mass conservation for Biot’s consolidation problem on the discrete level.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data Availability

Enquiries about data availability should be directed to the authors.

References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev spaces. Pure and Applied Mathematics, vol. 140, 2nd edn. Elsevier, Amsterdam (2003)

    MATH  Google Scholar 

  2. Anandarajah, A.: Computational methods in elasticity and plasticity. Solids and porous media, Springer, New York (2010)

    Book  MATH  Google Scholar 

  3. Barry, S.I., Mercer, G.N.: Exact solutions for two-dimensional time-dependent flow and deformation within a poroelastic medium. Trans. ASME J. Appl. Mech. 66(2), 536–540 (1999)

    Article  MathSciNet  Google Scholar 

  4. Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beirão da Veiga, L., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model. Numer. Anal. 51(2), 509–535 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beirão da Veiga, L., Lovadina, C., Vacca, G.: Virtual elements for the Navier-Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 56(3), 1210–1242 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berger, L., Bordas, R., Kay, D., Tavener, S.: Stabilized lowest-order finite element approximation for linear three-field poroelasticity. SIAM J. Sci. Comput. 37(5), A2222–A2245 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26(2), 182–185 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  9. Boffi, D., Botti, M., Di Pietro, D.A.: A nonconforming high-order method for the Biot problem on general meshes. SIAM J. Sci. Comput. 38(3), A1508–A1537 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  11. Boon, W.M., Kuchta, M., Mardal, K.-A., Ruiz-Baier, R.: Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot’s equations utilizing total pressure. SIAM J. Sci. Comput. 43(4), B961–B983 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  13. Brenner, S.C., Sung, L.-Y.: Virtual element methods on meshes with small edges or faces. Math. Models Methods Appl. Sci. 28(7), 1291–1336 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brezzi, F., Falk, R.S., Marini, L.D.: Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. 48(4), 1227–1240 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bürger, R., Kumar, S., Mora, D., Ruiz-Baier, R., Verma, N.: Virtual element methods for the three-field formulation of time-dependent linear poroelasticity. Adv. Comput. Math. 47(1), 37 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cangiani, A., Georgoulis, E.H., Pryer, T., Sutton, O.J.: A posteriori error estimates for the virtual element method. Numer. Math. 137(4), 857–893 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317–1354 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Chen, L., Huang, J.: Some error analysis on virtual element methods. Calcolo 55(1), 23 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, L., Wang, F.: A divergence free weak virtual element method for the Stokes problem on polytopal meshes. J. Sci. Comput. 78(2), 864–886 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, Y., Luo, Y., Feng, M.: Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem. Appl. Math. Comput. 219(17), 9043–9056 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Coulet, J., Faille, I., Girault, V., Guy, N., Nataf, F.: A fully coupled scheme using virtual element method and finite volume for poroelasticity. Comput. Geosci. 24(2), 381–403 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. da Veiga, LBa., Brezzi, F., Marini, L.D., Russo, A.: \(H({{\rm div}})\) and \(H({{\bf curl}})\)-conforming virtual element methods. Numer. Math. 133(2), 303–332 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Di Pietro, D.A., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fu, G.: A high-order HDG method for the Biot’s consolidation model. Comput. Math. Appl. 77(1), 237–252 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guo, J., Feng, M.: A new projection-based stabilized virtual element method for the Stokes problem. J. Sci. Comput. 85(1), 28 (2020)

    Article  MathSciNet  Google Scholar 

  26. Hong, Q., Kraus, J., Lymbery, M., Philo, F.: Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelasticity models. Numer. Linear Algebra Appl. 26(4), e2242 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hu, X., Rodrigo, C., Gaspar, F.J., Zikatanov, L.T.: A nonconforming finite element method for the Biot’s consolidation model in poroelasticity. J. Comput. Appl. Math. 310, 143–154 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kanschat, G., Riviere, B.: A finite element method with strong mass conservation for Biot’s linear consolidation model. J. Sci. Comput. 77(3), 1762–1779 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Korsawe, J., Starke, G.: A least-squares mixed finite element method for Biot’s consolidation problem in porous media. SIAM J. Numer. Anal. 43(1), 318–339 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kumar, S., Oyarzúa, R., Ruiz-Baier, R., Sandilya, R.: Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity. ESAIM Math. Model. Numer. Anal. 54(1), 273–299 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lee, J.J.: Robust three-field finite element methods for Biot’s consolidation model in poroelasticity. BIT 58(2), 347–372 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lee, J.J., Piersanti, E., Mardal, K.-A., Rognes, M.E.: A mixed finite element method for nearly incompressible multiple-network poroelasticity. SIAM J. Sci. Comput. 41(2), A722–A747 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mattheij, R., Molenaar, J.: Ordinary differential equations in theory and practice. Classics in Applied Mathematics, vol. 43. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002). (Reprint of the 1996 original)

  35. Murad, M.A., Loula, A.F.D.: On stability and convergence of finite element approximations of Biot’s consolidation problem. Int. J. Numer. Methods Eng. 37(4), 645–667 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Murad, M.A., Thomée, V., Loula, A.F.D.: Asymptotic behavior of semidiscrete finite-element approximations of Biot’s consolidation problem. SIAM J. Numer. Anal. 33(3), 1065–1083 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Oyarzúa, R., Ruiz-Baier, R.: Locking-free finite element methods for poroelasticity. SIAM J. Numer. Anal. 54(5), 2951–2973 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. I. The continuous in time case. Comput. Geosci. 11(2), 131–144 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Phillips, P.J., Wheeler, M.F.: A coupling of mixed and discontinuous Galerkin finite element methods for poroelasticity. Comput. Geosci. 12(4), 417–435 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rodrigo, C., Gaspar, F.J., Hu, X., Zikatanov, L.T.: Stability and monotonicity for some discretizations of the Biot’s consolidation model. Comput. Methods Appl. Mech. Eng. 298, 183–204 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Showalter, R.E.: Diffusion in poro-elastic media. J. Math. Anal. Appl. 251(1), 310–340 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sun, M., Rui, H.: A coupling of weak Galerkin and mixed finite element methods for poroelasticity. Comput. Math. Appl. 73(5), 804–823 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tang, X., Liu, Z., Zhang, B., Feng, M.: On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity. ESAIM Math. Model. Numer. Anal. 55(suppl.), S909–S939 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  44. Terzaghi, K.: Theoretical Soil Mechanics. Chapman, Orange (2007)

    Google Scholar 

  45. Ženíšek, A.: The existence and uniqueness theorem in Biot’s consolidation theory. Apl. Mat. 29(3), 194–211 (1984)

    MathSciNet  MATH  Google Scholar 

  46. Wang, G., Wang, F., Chen, L., He, Y.: A divergence free weak virtual element method for the Stokes–Darcy problem on general meshes. Comput. Methods Appl. Mech. Eng. 344, 998–1020 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  47. Wang, J., Wang, R., Zhai, Q., Zhang, R.: A systematic study on weak Galerkin finite element methods for second order elliptic problems. J. Sci. Comput. 74(3), 1369–1396 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  48. Yi, S.-Y.: A study of two modes of locking in poroelasticity. SIAM J. Numer. Anal. 55(4), 1915–1936 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zeng, Y., Cai, M., Wang, F.: An \({{\rm H}}{{\rm (div)}}\)-conforming finite element method for the Biot consolidation model. East Asian J. Appl. Math. 9(3), 558–579 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to Editors and anonymous reviewers for their valuable suggestions on an earlier version.

Funding

Research supported by the National Natural Science Foundation of China (No.11971337) and the Scientific Research Foundation of Chengdu University of Information Technology (KYTZ202184).

Author information

Authors and Affiliations

  1. College of Mathematics, Sichuan University, Chengdu, 610064, People’s Republic of China

    Jun Guo & Minfu Feng

  2. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, People’s Republic of China

    Jun Guo

Authors
  1. Jun Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Minfu Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minfu Feng.

Ethics declarations

Competing interests

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, J., Feng, M. A Robust and Mass Conservative Virtual Element Method for Linear Three-field Poroelasticity. J Sci Comput 92, 95 (2022). https://doi.org/10.1007/s10915-022-01960-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-01960-2

Keywords

Search

Navigation