Abstract
In this paper we consider a three-field formulation of the Biot model which has the displacement, the total pressure, and the pore pressure as unknowns. For parameter-robust stability analysis, we first show a priori estimates of the continuous problem with parameter-dependent norms. Then we study finite element discretizations which provide parameter-robust error estimates and preconditioners. For finite element discretizations we consider standard mixed finite element as well as stabilized methods for the Stokes equations, and the complete error analysis of semidiscrete solutions is given. Abstract forms of parameter-robust preconditioners are investigated by the operator preconditioning approach. The theoretical results are illustrated with numerical experiments.
Similar content being viewed by others
References
Anandarajah, A.: Computational Methods in Elasticity and Plasticity: Solids and Porous Media, SpringerLink: Bücher. Springer, New York (2010)
Bærland, T., Lee, J.J., Mardal, K.-A., Winther, R.: Weakly imposed symmetry and robust preconditioners for Biot’s consolidation model. Comput. Methods Appl. Math. 17(3), 377–396 (2017)
Bause, M., Radu, F.A., Köcher, U.: Space-time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods Appl. Mech. Eng. 320, 745–768 (2017)
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)
Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182–185 (1955)
Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations, Efficient solutions of elliptic systems (Kiel, 1984), Notes Numer. Fluid Mech., vol. 10, Friedr. Vieweg, Braunschweig, pp. 11–19. MR 804083 (1984)
Chaabane, N., Rivière, B.: A sequential discontinuous Galerkin method for the coupling of flow and geomechanics. J. Sci. Comput. 74(1), 375–395 (2018)
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)
Cheng, A.H.-D.: Poroelasticity, Theory and Applications of Transport in Porous Media, vol. 27. Springer, Cham (2016)
Falgout, R.D., Yang, U.M.: hypre: A library of high performance preconditioners. In: P.M.A. Sloot, A.G. Hoekstra, C.J. Kenneth Tan, J.J. Dongarra (Eds.) Computational Science—ICCS 2002 (Berlin, Heidelberg), pp. 632–641. Springer, Berlin (2002)
Feng, X., Ge, Z., Li, Y.: Analysis of a multiphysics finite element method for a poroelasticity model. IMA J. Numer. Anal. 38(1), 330–359 (2018)
Franca, L.P., Hughes, T.J.R., Stenberg, R.: Stabilized Finite Element Methods, Incompressible Computational Fluid Dynamics: Trends and Advances, pp. 87–107. Cambridge Univ. Press, Cambridge (2008)
Guosheng, F.: A high-order HDG method for the Biot’s consolidation model. Comput. Math. Appl. 77(1), 237–252 (2019)
Gaspar, F.J., Rodrigo, C., Hu, X., Ohm, P., Adler, J., Zikatanov, L.: New Stabilized Discretizations for Poroelasticity Equations, Numerical methods and applications, Lecture Notes in Comput. Sci., vol. 11189, pp. 3–14. Springer, Cham (2019)
Girault, V., Pierre-Arnaud, R.: Finite element methods for Navier–Stokes equations, Springer Series in Computational Mathematics, Theory and algorithms. vol. 5. Springer, Berlin (1986). MR 851383 (88b:65129)
Hong, Q., Kraus, J.: Parameter-robust stability of classical three-field formulation of Biot’s consolidation model. Electron. Trans. Numer. Anal. 48, 202–226 (2018)
Xiaozhe, H., 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)
Kechkar, N., Silvester, D.: Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58(197), 1–10 (1992)
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)
Jeonghun, J.: Lee, Robust error analysis of coupled mixed methods for Biot’s consolidation model. J. Sci. Comput. 69(2), 610–632 (2016)
Lee, J.J.: Robust three-field finite element methods for Biot’s consolidation model in poroelasticity. BIT 58(2), 347–372 (2018)
Jeonghun, J.: Lee, Kent-Andre Mardal, and Ragnar Winther, Parameter-robust discretization and preconditioning of Biot’s consolidation model. SIAM J. Sci. Comp. 39(1), A1–A24 (2017)
Logg, A., Mardal, K.-A., Wells, G.N. (eds.): Automated solution of differential equations by the finite element method. In: Lecture Notes in Computational Science and Engineering, vol. 84. Springer, Heidelberg (2012). The FEniCS book. MR 3075806
Murad, M.A., Loula, A.F.D.: Improved accuracy in finite element analysis of Biot’s consolidation problem. Comput. Methods Appl. Mech. Eng. 95(3), 359–382 (1992)
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)
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)
Oyarzúa, R., Ruiz-Baier, R.: Locking-free finite element methods for poroelasticity. SIAM J. Numer. Anal. 54(5), 2951–2973 (2016)
Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite element case. Comput. Geosci. 11(2), 131–144 (2007)
Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. II. The discrete-in-time case. Comput. Geosci. 11(2), 145–158 (2007)
Piersanti, E., Rognes, M.E.: Supplementary Material (Code) for a Mixed Finite Element Method For Nearly Incompressible Multiple-Network Poroelasticity’ by J. J. Lee, E. Piersanti, K.-A. Mardal and M. E. Rognes (2018)
Showalter, R.E.: Diffusion in Poro-elastic Media. J. Math. Anal. Appl. 251(1), 310–340 (2000)
Vermeer, P.A., Verruijt, A.: An accuracy condition for consolidation by finite elements. Int. J. Numer. Analyt. Methods Geomech. 5(1), 1–14 (1981)
Yi, S.-Y.: A coupling of nonconforming and mixed finite element methods for Biot’s consolidation model. Numer. Methods Part. Differ. Equ. 29(5), 1749–1777 (2013)
Yosida, K.: Functional Analysis, 6th ed., Springer Classics in Mathematics. Springer (1980)
Zienkiewicz, O.C., Shiomi, T.: Dynamic behaviour of saturated porous media; The generalized Biot formulation and its numerical solution. Int. J. Numer. Analyt. Methods Geomech. 8(1), 71–96 (1984)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has no relevant financial or non-financial interests to disclose.
Additional information
Communicated by Marko Huhtanen.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lee, J.J. Analysis and preconditioning of parameter-robust finite element methods for Biot’s consolidation model. Bit Numer Math 63, 42 (2023). https://doi.org/10.1007/s10543-023-00983-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10543-023-00983-x