×

Convergence analysis of a new mixed finite element method for Biot’s consolidation model. (English) Zbl 1350.74024

Summary: In this article, we propose a mixed finite element method for the two-dimensional Biot’s consolidation model of poroelasticity. The new mixed formulation presented herein uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure. This method is based on coupling two mixed finite element methods for each subproblem: the standard mixed finite element method for the flow subproblem and the Hellinger-Reissner formulation for the mechanical subproblem. Optimal a priori error estimates are proved for both semidiscrete and fully discrete problems when the Raviart-Thomas space for the flow problem and the Arnold-Winther space for the elasticity problem are used. In particular, optimality in the stress, displacement, and pressure has been proved in \(L^\infty(L^2)\) when the constrained-specific storage coefficient \(c_0\) is strictly positive and in the weaker \(L^2(L^2)\) norm when \(c_0\) is nonnegative. We also present some of our numerical results.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Biot, General theory of three-dimensional consolidation, J Appl Phys 12 pp 155– (1941) · JFM 67.0837.01
[2] Lewis, The finite element method in the static and dynamic deformation and consolidation of porous media (1998) · Zbl 0935.74004
[3] Phillips, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. I. The continuous in time case, Comput Geosci 11 pp 131– (2007) · Zbl 1117.74015
[4] Phillips, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. II. The discrete-in-time case, Comput Geosci 11 pp 145– (2007) · Zbl 1117.74016
[5] Phillips, A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput Geosci 12 pp 417– (2008) · Zbl 1155.74048
[6] Sandhu, Finite element analysis of seepage in elastic media, J Eng Mech Div ASCE 95 pp 641– (1969)
[7] Reed, An investigation of numerical errors in the analysis of consolidation by finite elements, Int J Numer Anal Methods Geomech 8 pp 243– (1984) · Zbl 0536.73089
[8] Booker, An investigation of the stability of numerical solutions of biot’s equations of consolidation, Int J Solids Struct 11 pp 907– (1975) · Zbl 0311.73047
[9] Phillips, Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach, Comput Geosci 13 pp 5– (2009) · Zbl 1172.74017
[10] Murad, Asymptotic behavior of semidiscrete finite-element approximations of Biot’s consolidation problem, SIAM J Numer Anal 33 pp 1065– (1996) · Zbl 0854.76053
[11] Wan, Stabilized finite element methods for coupled geomechanics and multiphase flow, Ph.D. Thesis (2002)
[12] Liu, Discontinuous Galerkin finite element solution for poromechanics problem, Ph.D. Thesis (2004)
[13] Haga, On the causes of pressure oscillations in low-permeable and low-compressible porous media, Int J Numer Anal Methods Geomech 36 pp 1507– (2012)
[14] Yi, A coupling of nonconforming and mixed finite element methods for Biot’s consolidation model, Numer Methods Partial Differ Equ 29 pp 1749– (2013) · Zbl 1274.74455
[15] Hellinger, Die allgemeinen ansätze der mechanik der kontinua, Encyclopadie der Mathematischen Wissenschaften 30 pp 602– (1914) · JFM 45.1012.01
[16] Arnold, Mixed finite elements for elasticity, Numer Math 92 pp 401– (2002) · Zbl 1090.74051
[17] Arnold, Rectangular mixed finite elements for elasticity, Math Models Methods Appl Sci 15 pp 1417– (2005) · Zbl 1077.74044
[18] Korsawe, A least-squares mixed finite element method for biot’s consolidation problem in porous media, SIAM J Numer Anal 43 pp 318– (2005) · Zbl 1086.76041
[19] Tchonkova, A new mixed finite element method for poro-elasticity, Int J Numer Anal Meth Geomech 32 pp 579– (2008) · Zbl 1273.74550
[20] Brezzi, Springer Series in Computational Mathematics 15 (1991)
[21] Terzaghi, Principles of soil mechanics, Eng News Rec 95 (1925)
[22] Biot, Theory of elasticity and consolidation for a porous anisotropic solid, J Appl Phys 26 pp 182– (1955) · Zbl 0067.23603
[23] Showalter, Diffusion in poro-elastic media, J Math Anal Appl 251 (1) pp 310– (2000) · Zbl 0979.74018
[24] Showalter, Partially saturated flow in a poroelastic medium, Discrete and Continuous Dynamical Systems Series B 1 pp 403– (2001) · Zbl 1004.76090
[25] Brenan, Classics in Applied Mathematics 14 (1996)
[26] Horn, Matrix analysis (1985) · Zbl 0576.15001
[27] Mandel, Consolidation des sols (étude mathématique) Géotechnique 30 pp 287– (1953)
[28] Cheng, A direct boundary element method for plane strain poroelasticity, Int J Numer Anal Methods Geomecha 12 pp 551– (1988) · Zbl 0662.73056
[29] Chen, Conforming rectangular mixed finite elements for elasticity, J Sci Comput 47 pp 93– (2011) · Zbl 1248.74038
[30] Hairer, Springer Series in Computational Mathematics 14 (2010)
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.