Locking-free finite element methods for poroelasticity. (English) Zbl 1457.65210


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74B10 Linear elasticity with initial stresses
35B20 Perturbations in context of PDEs
35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
Full Text: DOI


[1] G. Aguilar, F. Gaspar, F. Lisbona, and C. Rodrigo, Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation, Internat. J. Numer. Methods Engrg., 75 (2008), pp. 1282–1300. · Zbl 1158.74473
[2] I. Babuška and A. Aziz, Survey lectures on the mathematical foundations of the finite element method, in the Mathematical Foundations of the Finite Element Method with Applications to PDEs, Academic Press, New York, 1972, pp. 1–359.
[3] S. Badia, A. Quaini, and A. Quarteroni, Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction, J. Comput. Phys., 228 (2009), pp. 7986–8014. · Zbl 1391.74234
[4] D. Baroli, A. Quarteroni, and R. Ruiz-Baier, Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity, Adv. Comput. Math., 39 (2013), pp. 425–443. · Zbl 1271.74030
[5] T. Barth, P. Bochev, M. Gunzburger, and J. Shadid, A taxonomy of consistently stabilized finite element methods for the Stokes problem, SIAM J. Sci. Comput., 25 (2004), pp. 1585–1607, doi:10.1137/S1064827502407718. · Zbl 1133.76307
[6] 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), pp. A2222–A2245, doi:10.1137/15M1009822. · Zbl 1326.76054
[7] M. A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26 (1955), pp. 182–185. · Zbl 0067.23603
[8] D. Boffi, Stability of higher order triangular Hood-Taylor methods for stationary Stokes equations, Math. Models Methods Appl. Sci., 2 (1994), pp. 223–235, doi:10.1142/S0218202594000133. · Zbl 0804.76051
[9] D. Boffi, Three-dimensional finite element methods for the Stokes problem, SIAM J. Numer. Anal., 34 (1997), pp. 664–670, doi:10.1137/S0036142994270193.
[10] Y. Chen, Y. Luo, and M. Feng, Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem, Appl. Math. Comput., 219 (2013), pp. 9043–9056. · Zbl 1290.74038
[11] C. Domínguez, G.N. Gatica, S. Meddahi, and R. Oyarzúa, A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem, ESAIM Math. Model. Numer. Anal., 47 (2013), pp. 471–506.
[12] V. Domíngues and F. J. Sayas, A BEM-FEM overlapping algorithm for the Stokes equation, Appl. Math. Comput., 182 (2006), pp. 691–710. · Zbl 1120.76047
[14] L. Franca, T. J. R. Hughes, and R. Stenberg, Stabilized Finite Element Methods for the Stokes Problem, Incompressible Computational Fluid Dynamics, M. Gunzburger and R.A. Nicolaides, eds., Cambridge University Press, Cambridge, UK, 1993, pp. 87–107. · Zbl 1189.76339
[15] F. J. Gaspar, F. J. Lisbona, and C. W. Oosterlee, A stabilized difference scheme for deformable porous media and its numerical resolution by multigrid methods, Comput. Vis. Sci., 11 (2008), pp. 67–76.
[16] G. N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications, Springer Briefs in Mathematics, Springer, Cham, 2014. · Zbl 1293.65152
[17] G. N. Gatica, R. Oyarzúa, and F. J. Sayas, Convergence of a family of Galerkin discretizations for the Stokes–Darcy coupled problem, Numer. Methods Partial Differential Equations, 27 (2011), pp. 721–748. · Zbl 1301.76046
[18] G. N. Gatica, A. Márquez, and S. Meddahi, Analysis of the coupling of primal and dual-mixed finite element methods for a two-dimensional fluid-solid interaction problem, SIAM J. Numer. Anal., 45 (2007), pp. 2072–2097, doi:10.1137/060660370. · Zbl 1225.74087
[19] G. N. Gatica, R. Oyarzúa, and F.J. Sayas, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem, Math. Comp., 80 (2011), pp. 1911–1948. · Zbl 1301.76047
[20] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier–Stokes Equations, Lecture Notes in Math. 749, Springer-Verlag, Berlin, New York, 1979. · Zbl 0413.65081
[21] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics Appl. Math. 69, SIAM, Philadelphia, 2011. · Zbl 1231.35002
[22] J. Korsawe and G. Starke, A least-squares mixed finite element method for Biot’s consolidation problem in porous media, SIAM J. Numer. Anal., 43 (2005), pp. 318–339, doi:10.1137/S0036142903432929. · Zbl 1086.76041
[23] J. J. Lee, Guaranteed Locking-Free Finite Element Methods for Biot’s Consolidation Model in Poroelasticity, http://arxiv.org/abs/1403.7038, 2015.
[24] R. Liu, M. F. Wheeler, C. N. Dawson, and R. H. Dean, On a coupled discontinuous/continuous Galerkin framework and an adaptive penalty scheme for poroelasticity problems, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 3499–3510, doi:10.1016/j.cma.2009.07.005. · Zbl 1230.74189
[25] M. A. Murad and A. F. D. Loula, On stability and convergence of finite element approximations of Biot’s consolidation problem, Internat. J. Numer. Methods Engrg., 37 (1994), pp. 645–667. · Zbl 0791.76047
[26] M. A. Murad, V. Thomée, and A. F. D. Loula, Asymptotic behavior of semidiscrete finite-element approximations of Biot’s consolidation problem, SIAM J. Numer. Anal., 33 (1996), pp. 1065–1083, doi:10.1137/0733052.
[27] P. J. Phillips, Finite Element Methods in Linear Poroelasticity: Theoretical and Computational Results, Ph.D. thesis, The University of Texas at Austin, Austin, TX, 2005.
[28] 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), pp. 131–144. · Zbl 1117.74015
[29] R. Ruiz-Baier and I. Lunati, Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models, J. Comput. Phys., 322 (2016), pp. 666-688, doi:10.1016/2016.06.054. · Zbl 1351.76079
[30] R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl., 251 (2000), pp. 310–340. · Zbl 0979.74018
[31] K. Strehlow, J. H. Gottsmann, and A. C. Rust, Poroelastic responses of confined aquifers to subsurface strain and their use for volcano monitoring, Solid Earth, 6 (2015), pp. 1207–1229.
[33] A. Truty, A Galerkin/least-squares finite element formulation for consolidation, Internat. J. Numer. Methods Engrg., 52 (2001), pp. 763–786. · Zbl 1017.74077
[34] R. Uzuoka and R. I. Borja, Dynamics of unsaturated poroelastic solids at finite strain, Int. J. Numer. Anal. Meth. Geomech., 36 (2012), pp. 1535–1573, ŭldoi: 10.1002/nag.1061.
[35] J. Wan, Stabilized Finite Element Method for Coupled Geomechanics and Multiphase Flow, Ph.D. thesis, Stanford University, Stanford, CA, 2002.
[36] M. F. Wheeler, G. Xue, and I. Yotov, Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity, Comput. Geosci., 18 (2014), pp. 57–75. · Zbl 1395.65093
[37] J. A. White and R. I. Borja, Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 4353–4366. · Zbl 1194.74480
[38] S.-Y. Yi, Convergence analysis of a new mixed finite element method for Biot’s consolidation model, Numer. Methods Partial Differential Equations, 30 (2014), pp. 1189–1210. · Zbl 1350.74024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.