no. Supplément
On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S909-S939

We propose and analyze two locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity. One is a high-order scheme, and the other is a low-order scheme. For time discretization, we use the backward Euler scheme. The proposed methods are well-posed, and optimal error estimates of all the unknowns are obtained for fully discrete solutions. The generic constants in the estimates are uniformly bounded as the Lamé coefficient λ tends to infinity, and as the constrained specific storage coefficient is arbitrarily small. Therefore the methods are free of both Poisson locking and pressure oscillations. Numerical results illustrate the good performance of the methods and confirm our theoretical predictions.

DOI : 10.1051/m2an/2020064
Classification : 65M12, 65M60
Keywords: Virtual elements, poroelasticity, Biot’s consolidation model, locking-free, pressure oscillations 
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     title = {On the locking-free three-field virtual element methods for {Biot{\textquoteright}s} consolidation model in poroelasticity},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {S909--S939},
     year = {2021},
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Tang, Xialan; Liu, Zhibin; Zhang, Baiju; Feng, Minfu. On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S909-S939. doi: 10.1051/m2an/2020064

[1] R. Adams and J. Fournier, Sobolev Spaces. Vol. 140 of Pure and Applied Mathematics. Academic Press (2003). | MR | Zbl

[2] B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013) 376–391. | MR

[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. | MR | Zbl

[4] P. F. Antonietti, L. Beirão Da Veiga, S. Scacchi and M. Verani, A C 1 virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54 (2016) 34–56. | MR

[5] E. Artioli, S. De Miranda, C. Lovadina and L. Patruno, A stress/displacement virtual element method for plane elasticity problems. Comput. Methods Appl. Mech. Eng. 325 (2017) 155–174. | MR

[6] S. I. Barry and G. N. Mercer, Exact solutions for two-dimensional time-dependent flow and deformation within a poroelastic medium. Trans. ASME J. Appl. Mech. 6 (1999) 536–540. | MR

[7] L. Beirão Da Veiga and K. Lipnikov, A mimetic discretization of the Stokes problem with selected edge bubbles. SIAM J. Sci. Comput. 32 (2010) 875–893. | MR | Zbl

[8] L. Beirão Da Veiga, F. Brezzi and L. D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812. | MR | Zbl

[9] L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | MR | Zbl

[10] L. Beirão Da Veiga, F. Brezzi, L. D. Marini and A. Russo, The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24 (2014) 1541–1573. | MR | Zbl

[11] L. Beirão Da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method for Elliptic Problems. Vol. 11 of MS&A – Modeling, Simulation and Applications. Springer, Cham (2014). | MR | Zbl

[12] L. Beirão Da Veiga, F. Brezzi, L. D. Marini and A. Russo, Virtual Element Implementation for General Elliptic Equations. Springer International Publishing, Cham (2016) 39–71. | MR

[13] L. Beirão Da Veiga, F. Brezzi, L. D. Marini and A. Russo, Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26 (2016) 729–750. | MR

[14] 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: M2AN 50 (2016) 727–747. | MR | Zbl | Numdam

[15] L. Beirão Da Veiga, C. Lovadina and A. Russo, Stability analysis for the virtual element method. Math. Models Methods Appl. Sci. 27 (2017) 2557–2594. | MR

[16] L. Beirão Da Veiga, C. Lovadina and G. Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM: M2AN 51 (2017) 509–535. | MR | Zbl | Numdam

[17] L. Beirão Da Veiga, C. Lovadina and G. Vacca, Virtual elements for the Navier-Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 56 (2018) 1210–1242. | MR

[18] M. F. Benedetto, S. Berrone, A. Borio, S. Pieraccini and S. Scialò, Order preserving SUPG stabilization for the virtual element formulation of advection–diffusion problems. Comput. Methods Appl. Mech. Eng. 311 (2016) 18–40. | MR

[19] L. Berger, R. Bordas, D. Kay and S. Tavener, Stabilized lowest-order finite element approximation for linear three-field poroelasticity. SIAM J. Sci. Comput. 37 (2015) A2222–A2245. | MR

[20] C. Bernardi and G. Raugel, Analysis of some finite elements for the Stokes problem. Math. Comput. 44 (1985) 71–79. | MR | Zbl

[21] M. A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12 (1941) 155–164. | JFM

[22] M. A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26 (1955) 182–185. | MR | Zbl

[23] D. Boffi, M. Botti and D. A. Di Pietro, A nonconforming high-order method for the Biot problem on general meshes. SIAM J. Sci. Comput. 38 (2016) A1508–A1537. | MR

[24] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods. Vol. 15 of Texts in Applied Mathematics. Springer, New York (2007). | MR | Zbl

[25] S. C. Brenner, Q. Guan and L. Sung, Some estimates for virtual element methods. Comput. Methods Appl. Math. 17 (2017) 553–574. | MR

[26] F. Brezzi, R. S. Falk and L. D. Marini, Basic principles of mixed virtual element methods. ESAIM: M2AN 48 (2014) 1227–1240. | MR | Zbl | Numdam

[27] R. Bürger, S. Kumar, D. Mora, R. Ruiz-Baier and N. Verma, Virtual element methods for the three-field formulation of time–dependent linear poroelasticity. Preprint . | arXiv | MR

[28] A. Cangiani, E. H. Georgoulis, T. Pryer and O. J. Sutton, A posteriori estimates for the virtual element method. Numer. Math. 137 (2017) 857–893. | MR

[29] O. Čertík, F. Gardini, G. Manzini and G. Vacca, The virtual element method for eigenvalue problems with potential terms on polytopic meshes. Appl. Math. 63 (2018) 333–365. | MR

[30] G. Chen and M. Feng, Stabilized finite element methods for Biot’s consolidation problems using equal order elements. Adv. Appl. Math. Mech. 10 (2018) 77–99. | MR

[31] L. Chen and J. Huang, Some error analysis on virtual element methods. Calcolo 55 (2018) 5. | MR

[32] Y. Chen, Y. Luo and M. Feng, Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem. Appl. Math. Comput. 219 (2013) 9043–9056. | MR | Zbl

[33] Y. Chen, G. Chen and X. Xie, Weak Galerkin finite element method for Biot’s consolidation problem. J. Comput. Appl. Math. 330 (2018) 398–416. | MR

[34] J. Coulet, I. Faille, V. Girault, N. Guy and F. Nataf, Fully coupled schemes using virtual element and finite volume discretisations for Biot equations modelling. In: ECMOR XVI–16th European Conference on the Mathematics of Oil Recovery (2018).

[35] M. Fortin, Old and new finite elements for incompressible flows. Int. J. Numer. Methods Fluids 1 (1981) 347–364. | MR | Zbl

[36] G. Fu, A high-order HDG method for the Biot’s consolidation model. Comput. Math. Appl. 77 (2019) 237–252. | MR

[37] A. L. Gain, C. Talischi and G. H. Paulino, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput. Methods Appl. Mech. Eng. 282 (2014) 132–160. | MR

[38] F. Gardini and G. Vacca, Virtual element method for second-order elliptic eigenvalue problems. IMA J. Numer. Anal. 38 (2018) 2026–2054. | MR

[39] F. Gardini, G. Manzini and G. Vacca, The nonconforming Virtual Element Method for eigenvalue problems. ESAIM: M2AN 53 (2019) 749–774. | MR | Zbl | Numdam

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

[41] X. Hu, L. Mu and X. Ye, Weak Galerkin method for the Biot’s consolidation model. Comput. Math. Appl. 75 (2018) 2017–2030. | MR | DOI

[42] J. J. Lee, Robust three-field finite element methods for Biot’s consolidation model in poroelasticity. BIT 58 (2018) 347–372. | MR | DOI

[43] X. Liu and Z. Chen, The nonconforming virtual element method for the Navier-Stokes equations. Adv. Comput. Math. 45 (2019) 51–74. | MR | DOI

[44] D. Mora, G. Rivera and R. Rodríguez, A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25 (2015) 1421–1445. | MR | DOI

[45] M. Murad and A. Loula, Improved accuracy in finite element analysis of Biot’s consolidation problem. Comput. Methods Appl. Mech. Eng. 95 (1992) 359–382. | MR | Zbl | DOI

[46] M. Murad and A. Loula, On stability and convergence of finite element approximations of Biot’s consolidation problem. Int. J. Numer. Methods Eng. 37 (1994) 645–667. | MR | Zbl | DOI

[47] M. Murad, V. Thomée and A. Loula, Asymptotic behavior of semidiscrete finite-element approximations of Biot’s consolidation problem. SIAM J. Numer. Anal. 33 (1996) 1065–1083. | MR | Zbl | DOI

[48] P. J. Phillips, Finite element methods in linear poroelasticity: theoretical and computational results, Ph.D. thesis. ProQuest LLC, Ann Arbor, MI/The University of Texas, Austin (2005). | MR

[49] P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case. Comput. Geosci. 11 (2007) 131–144. | MR | Zbl | DOI

[50] P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete in time case. Comput. Geosci. 11 (2007) 145–158. | MR | Zbl | DOI

[51] P. J. Phillips and M. F. Wheeler, A coupling of mixed and discontinuous Galerkin finite element methods for poroelasticity. Comput. Geosci. 12 (2008) 417–435. | MR | Zbl | DOI

[52] P. J. Phillips and M. F. Wheeler, Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput. Geosci. 13 (2009) 5–12. | Zbl | DOI

[53] M. B. Reed, An investigation of numerical errors in the analysis of consolidation by finite elements. Int. J. Numer. Anal. 8 (1984) 234–257. | Zbl

[54] C. Rodrigo, F. Gaspar, X. Hu and L. Zikatanov, Stability and monotonicity for some discretizations of the Biot’s consolidation model. Comput. Methods Appl. Mech. Eng. 298 (2016) 183–204. | MR | Zbl | DOI

[55] O. J. Sutton, The virtual element method in 50 lines of MATLAB. Numer. Algorithms 75 (2016) 1141–1159. | MR | Zbl | DOI

[56] X. Tang, Z. Liu, B. Zhang and M. Feng, A low-order locking-free virtual element for linear elasticity problems. Comput. Math. Appl. 80 (2020) 1260–1274. | MR | Zbl | DOI

[57] K. Terzaghi, Theoretical Soil Mechanics. John Wiley & Sons, New York (1943). | DOI

[58] G. Vacca, Virtual element methods for hyperbolic problems on polygonal meshes. Comput. Math. Appl. 74 (2017) 882–898. | MR | Zbl | DOI

[59] G. Vacca and L. Beirão Da Veiga, Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Part. Differ. Equ. 31 (2015) 2110–2134. | MR | Zbl | DOI

[60] P. Vermeer and A. Verruijt, An accuracy condition for consolidation by finite elements. Int. J. Numer. Anal. Methods Geomech. 5 (1981) 1–14. | MR | Zbl | DOI

[61] S.-Y. Yi, A coupling of nonconforming and mixed finite element methods for Biot’s consolidation model. Numer. Methods Part. Differ. Equ. 29 (2013) 1749–1777. | MR | Zbl | DOI

[62] S.-Y. Yi, A study of two modes of locking in poroelasticity. SIAM J. Numer. Anal. 55 (2017) 1915–1936. | MR | Zbl | DOI

[63] B. Zhang and M. Feng, Virtual element method for two-dimensional linear elasticity problem in mixed weakly symmetric formulation. Appl. Math. Comput. 328 (2018) 1–25. | MR | Zbl

[64] B. Zhang, Y. Yang and M. Feng, Mixed virtual element methods for elastodynamics with weak symmetry. J. Comput. Appl. Math. 353 (2018) 49–71. | MR | Zbl | DOI

[65] O. C. Zienkiewicz and T. Shiomi, Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution. Int. J. Numer. Anal. Methods Geomech. 8 (1984) 71–96. | Zbl | DOI

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