no. Supplément
Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S475-S506

We present an a priori and a posteriori error analysis of a conforming finite element method for a four-field formulation of the steady-state Biot’s consolidation model. For the a priori error analysis we provide suitable hypotheses on the corresponding finite dimensional subspaces ensuring that the associated Galerkin scheme is well-posed. We show that a suitable choice of subspaces is given by the Raviart–Thomas elements of order k ≥ 0 for the fluid flux, discontinuous polynomials of degree k for the fluid pressure, and any stable pair of Stokes elements for the solid displacements and total pressure. Next, we develop a reliable and efficient residual-based a posteriori error estimator. Both the reliability and efficiency estimates are shown to be independent of the modulus of dilatation. Numerical examples in 2D and 3D verify our analysis and illustrate the performance of the proposed a posteriori error indicator.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1051/m2an/2020045
Classification : 65N30, 76S05, 74F10, 65N15
Keywords: Biot model, mixed finite element methods, locking free approximations, $$ error analysis 
@article{M2AN_2021__55_S1_S475_0,
     author = {Oyarz\'ua, Ricardo and Rhebergen, Sander and Solano, Manuel and Z\'u\~niga, Paulo},
     title = {Error analysis of a conforming and locking-free four-field formulation for the stationary {Biot{\textquoteright}s} model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {S475--S506},
     year = {2021},
     publisher = {EDP-Sciences},
     volume = {55},
     number = {Suppl\'ement},
     doi = {10.1051/m2an/2020045},
     mrnumber = {4221320},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2020045/}
}
TY  - JOUR
AU  - Oyarzúa, Ricardo
AU  - Rhebergen, Sander
AU  - Solano, Manuel
AU  - Zúñiga, Paulo
TI  - Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2021
SP  - S475
EP  - S506
VL  - 55
IS  - Supplément
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2020045/
DO  - 10.1051/m2an/2020045
LA  - en
ID  - M2AN_2021__55_S1_S475_0
ER  - 
%0 Journal Article
%A Oyarzúa, Ricardo
%A Rhebergen, Sander
%A Solano, Manuel
%A Zúñiga, Paulo
%T Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2021
%P S475-S506
%V 55
%N Supplément
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2020045/
%R 10.1051/m2an/2020045
%G en
%F M2AN_2021__55_S1_S475_0
Oyarzúa, Ricardo; Rhebergen, Sander; Solano, Manuel; Zúñiga, Paulo. Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S475-S506. doi: 10.1051/m2an/2020045

[1] E. Ahmed, F. A. Radu and J. M. Nordbotten, Adaptive poromechanics computations based on a posteriori error estimates for fully mixed formulations of Biot’s consolidation model. Comput. Methods Appl. Mech. Eng. 347 (2019) 264–294. | MR | DOI

[2] E. Ahmed, J. M. Nordbotten and F. A. Radu, Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems. J. Comput. Appl. Math. 364 (2020) 112312. | MR | DOI

[3] M. S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N. Wells, The fenics project version 1.5. Arch. Numer. Softw. 3 (2015) 9–23.

[4] M. Alvarez, G. N. Gatica and R. Ruiz-Baier, A posteriori error analysis for a viscous flow-transport problem. ESAIM: M2AN 50 (2016) 1789–1816. | MR | Zbl | Numdam | DOI

[5] P. R. Amestoy, I. S. Duff and J.-Y. L’Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184 (2000) 501–520. | Zbl | DOI

[6] R. Araya, M. Solano and P. Vega, Analysis of an adaptive HDG method for the Brinkman problem. IMA J. Numer. Anal. 39 (2019) 1502–1528. | MR | DOI

[7] D. N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. | MR | Zbl | DOI

[8] D. N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984) 337–344. | MR | Zbl | DOI

[9] I. Babuška and M. Suri, Locking effects in the finite element approximation of elasticity problems. Numer. Math. 62 (1992) 439–463. | MR | Zbl | DOI

[10] M. Bause, F. A. Radu and U. Köcher, Space-time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods Appl. Mech. Eng. 320 (2017) 745–768. | MR | DOI

[11] P. J. Basser, Interstitial pressure, volume, and flow during infusion into brain tissue. Microvasc. Res. 44 (1992) 143–165. | DOI

[12] 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 | DOI

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

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

[15] D. Boffi, Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34 (1997) 664–670. | MR | Zbl | DOI

[16] D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications, Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1. Edited by Boffi and Lucia Gastaldi. Lecture Notes in Mathematics, 1939. Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence (2008). | MR | Zbl

[17] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications. In: Vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013). | MR | Zbl

[18] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). | MR | Zbl

[19] F. Brezzi and R. S. Falk, Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28 (1991) 581–590. | MR | Zbl | DOI

[20] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. In: Vol. 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York (1991). | MR | Zbl

[21] C. Carstensen, A posteriori error estimate for the mixed finite element method. Math. Comput. 66 (1997) 465–476. | MR | Zbl

[22] C. Carstensen and G. Dolzmann, A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81 (1998) 187–209. | MR | Zbl

[23] S. Caucao, D. Mora and R. Oyarzúa, A priori and a posteriori error analysis of a pseudostress-based mixed formulation of the Stokes problem with varying density. IMA J. Numer. Anal. 36 (2016) 947–983. | MR

[24] 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

[25] Z. Chen, Y. Xu and J. Zhang, A second-order hybrid finite volume method for solving the Stokes equation. Appl. Numer. Math. 119 (2017) 213–224. | MR

[26] P. G. Ciarlet, The Finite Element Method for Elliptic Problems. In: Vol. 4 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam-New York-Oxford (1978). | MR | Zbl

[27] P. Clément, Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975) 77–84. | MR | Zbl | Numdam

[28] C. Domnguez, G. N. Gatica and S. Meddahi, A posteriori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem. J. Comput. Math. 33 (2015) 606–641. | MR

[29] A. Ern and S. Meunier, A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems. ESAIM: M2AN 43 (2009) 353–375. | MR | Zbl | Numdam

[30] Q. Fang, Mesh-based monte carlo method using fast ray-tracing in plücker coordinates. Biomed. Opt. Express 1 (2010) 165–175. | DOI

[31] Z. E. A. Fellah, N. Sebaa, M. Fellah, E. Ogam, F. G. Mitri, C. Depollier and W. Lauriks, Application of the Biot model to ultrasound in bone: inverse problem. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55 (2008) 1516–1523. | DOI

[32] 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. Visual. Sci. 11 (2008) 67–76. | MR | DOI

[33] G. N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014). | MR | Zbl

[34] G. N. Gatica, A note on stable helmholtz decompositions in 3D. Appl. Anal. 99 (2018) 1110–1121. | MR | DOI

[35] G. N. Gatica, A. Márquez and M. A. Sánchez, Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Eng. 199 (2010) 1064–1079. | MR | Zbl | DOI

[36] G. N. Gatica, R. Oyarzúa and F.-J. Sayas, A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes-Darcy coupled problem. Comput. Methods Appl. Mech. Eng. 200 (2011) 1877–1891. | MR | Zbl | DOI

[37] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. In: Vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). | MR | Zbl | DOI

[38] L. Guo, J. C. Vardakis, T. Lassila, M. Mitolo, N. Ravikumar, D. Chou, M. Lange, A. Sarrami-Foroushani, B. J. Tully, Z. A. Taylor and S. Varma, Subject-specific multi-poroelastic model for exploring the risk factors associated with the early stages of Alzheimer’s disease. Interface Focus 8 (2017) 20170019. | DOI

[39] G. Jayaraman, Water transport in the arterial wall – a theoretical study. J. Biomech. 16 (1983) 833–840. | DOI

[40] J.-M. Kim and R. R. Parizek, Three-dimensional finite element modelling for consolidation due to groundwater withdrawal in a desaturating anisotropic aquifer system. Int. J. Numer. Anal. Methods Geomech. 23 (1999) 549–571. | Zbl | DOI

[41] S. Kumar, R. Oyarzúa, R. Ruiz-Baier and R. Sandilya, Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity. ESAIM: M2AN 54 (2020) 273–299. | MR | Numdam | DOI

[42] J. J. Lee, K.-A. Mardal and R. Winther, Parameter-robust discretization and preconditioning of Biot’s consolidation model. SIAM J. Sci. Comput. 39 (2017) A1–A24. | MR | DOI

[43] J. J. Lee, E. Piersanti, K.-A. Mardal and M. E. Rognes, A mixed finite element method for nearly incompressible multiple-network poroelasticity. SIAM J. Sci. Comput. 41 (2019) A722–A747. | MR | DOI

[44] X. Li, H. Von Holst and S. Kleiven, Influences of brain tissue poroelastic constants on intracranial pressure (ICP) during constant-rate infusion. Comput. Methods Biomech. Biomed. Eng. 16 (2013) 1330–1343. | DOI

[45] M. A. Murad and A. F. D. 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

[46] R. Oyarzúa and R. Ruiz-Baier, Locking-free finite element methods for poroelasticity. SIAM J. Numer. Anal. 54 (2016) 2951–2973. | MR | DOI

[47] 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

[48] 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

[49] A. Plaza and G. F. Carey, Local refinement of simplicial grids based on the skeleton. Appl. Numer. Math. 32 (2000) 195–218. | MR | Zbl | DOI

[50] R. Riedlbeck, D. A. Di Pietro, A. Ern, S. Granet and K. Kazymyrenko, Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis. Comput. Math. Appl. 73 (2017) 1593–1610. | MR | Zbl | DOI

[51] B. Rivière, J. Tan and T. Thompson, Error analysis of primal discontinuous Galerkin methods for a mixed formulation of the Biot equations. Comput. Math. Appl. 73 (2017) 666–683. | MR | Zbl | DOI

[52] R. E. Showalter, Diffusion in poro-elastic media. J. Math. Anal. Appl. 251 (2000) 310–340. | MR | Zbl | DOI

[53] R. E. Showalter, Diffusion in deformable media. In: Vol. 131 of Resource Recovery, Confinement, and Remediation of Environmental Hazards (Minneapolis, MN, 2000). The IMA Volumes in Mathematics and its Applications. Springer, New York (2002) 115–129. | MR | Zbl

[54] K. Terzaghi, Principle of soil mechanics. Engineering News Record, A Series of Articles (1925).

[55] B. Tully and Y. Ventikos, Cerebral water transport using multiple-network poroelastic theory: application to normal pressure hydrocephalus. J. Fluid Mech. 667 (2011) 188–215. | MR | Zbl | DOI

[56] J. C. Vardakis, D. Chou, B. J. Tully, C. C. Hung, T. H. Lee, P.-H. Tsui and Y. Ventikos, Investigating cerebral oedema using poroelasticity. Micro and Nano Flows 2014 (MNF2014) – Biomedical Stream. Med. Eng. Phys. 38 (2016) 48–57.

[57] R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) 67–83. | MR | Zbl | DOI

[58] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley Teubner, Chichester (1996). | Zbl

[59] R. Verfürth, A review of a posteriori error estimation techniques for elasticity problems. Comput. Methods Appl. Mech. Eng. 176 (1999) 419–440. | MR | Zbl | DOI

[60] M. Wangen, S. Gasda and T. Bjørnarå, Geomechanical consequences of large-scale fluid storage in the Utsira Formation in the North Sea. Energy Proc. 97 (2016) 486–493. | DOI

[61] M. Wheeler, G. Xue and I. Yotov, Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity. Comput. Geosci. 18 (2014) 57–75. | MR | Zbl | DOI

[62] 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. Eng. 197 (2008) 4353–4366. | Zbl | DOI

[63] 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

[64] S.-Y. Yi, Convergence analysis of a new mixed finite element method for Biot’s consolidation model. Numer. Methods Part. Differ. Equ. 30 (2014) 1189–1210. | MR | Zbl | DOI

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

[66] J. Young, B. Rivière, C. S. Cox, Jr. and K. Uray, A mathematical model of intestinal oedema formation. Math. Med. Biol. 31 (2014) 1–15. | MR | Zbl | DOI

Cité par Sources :