## A nonconforming finite element method for the Biot’s consolidation model in poroelasticity.(English)Zbl 1381.76175

Summary: A stable finite element scheme that avoids pressure oscillations for a three-field Biot’s model in poroelasticity is considered. The involved variables are the displacements, fluid flux (Darcy velocity), and the pore pressure, and they are discretized by using the lowest possible approximation order: Crouzeix-Raviart finite elements for the displacements, lowest order Raviart-Thomas-Nédélec elements for the Darcy velocity, and piecewise constant approximation for the pressure. Mass-lumping technique is introduced for the Raviart-Thomas-Nédélec elements in order to eliminate the Darcy velocity and, therefore, reduce the computational cost. We show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass-lumping. Finally, numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the Raviart-Thomas-Nédélec elements is used.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76S05 Flows in porous media; filtration; seepage
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### References:

 [1] Terzaghi, K., Theoretical soil mechanics, (1943), Wiley New York [2] Biot, M. A., General theory of threedimensional consolidation, J. Appl. Phys., 12, 2, 155-164, (1941) · JFM 67.0837.01 [3] Biot, M. A., Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26, 2, 182-185, (1955) · Zbl 0067.23603 [4] Gaspar, F. J.; Lisbona, F. J.; Vabishchevich, P. N., A finite difference analysis of biot’s consolidation model, Appl. Numer. Math., 44, 4, 487-506, (2003), URL http://dx.doi.org/10.1016/S0168-9274(02)00190-3 · Zbl 1023.76032 [5] Ferronato, M.; Castelletto, N.; Gambolati, G., A fully coupled 3-d mixed finite element model of Biot consolidation, J. Comput. Phys., 229, 12, 4813-4830, (2010) · Zbl 1305.76055 [6] Haga, J. B.; Osnes, H.; Langtangen, H. P., On the causes of pressure oscillations in low-permeable and low-compressible porous media, Int. J. Numer. Anal. Methods Geomech., 36, 12, 1507-1522, (2012) [7] M. Favino, A. Grillo, R. Krause, A stability condition for the numerical simulation of poroelastic systems, in: C. Hellmich, B. Pichler, D. Adam (Eds.), Poromechanics V: Proceedings of the Fifth Biot Conference on Poromechanics, 2013, pp. 919-928. URL http://ascelibrary.org/doi/abs/10.1061/9780784412992.110. [8] Phillips, P.; Wheeler, M., Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach, Comput. Geosci., 13, 1, 5-12, (2009) · Zbl 1172.74017 [9] Murad, M. A.; Loula, A. F.D., Improved accuracy in finite element analysis of biot’s consolidation problem, Comput. Methods Appl. Mech. Engrg., 95, 3, 359-382, (1992), URL http://dx.doi.org/10.1016/0045-7825(92)90193-N · Zbl 0760.73068 [10] Murad, M. A.; Loula, A. F.D., On stability and convergence of finite element approximations of biot’s consolidation problem, Internat. J. Numer. Methods Engrg., 37, 4, 645-667, (1994), URL http://dx.doi.org/10.1002/nme.1620370407 · Zbl 0791.76047 [11] 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), URL http://dx.doi.org/10.1137/0733052 · Zbl 0854.76053 [12] Aguilar, G.; Gaspar, F.; Lisbona, F.; Rodrigo, C., Numerical stabilization of biot’s consolidation model by a perturbation on the flow equation, Internat. J. Numer. Methods Engrg., 75, 11, 1282-1300, (2008), URL http://dx.doi.org/10.1002/nme.2295 · Zbl 1158.74473 [13] Rodrigo, C.; Gaspar, F.; Hu, X.; Zikatanov, L., Stability and monotonicity for some discretizations of the biot’s consolidation model, Comput. Methods Appl. Mech. Engrg., 298, 183-204, (2016) [14] Arnold, D. N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 4, 337-344, (1984), URL http://dx.doi.org/10.1007/BF02576171 · Zbl 0593.76039 [15] 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) · Zbl 1086.76041 [16] Tchonkova, M.; Peters, J.; Sture, S., A new mixed finite element method for poro-elasticity, Int. J. Numer. Anal. Methods Geomech., 32, 6, 579-606, (2008), URL http://dx.doi.org/10.1002/nag.630 · Zbl 1273.74550 [17] Phillips, P.; Wheeler, M., A coupling of mixed and continuous Galerkin finite element methods for poroelasticity i: the continuous in time case, Comput. Geosci., 11, 2, 131-144, (2007), URL http://dx.doi.org/10.1007/s10596-007-9045-y · Zbl 1117.74015 [18] Phillips, P.; Wheeler, M., 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), URL http://dx.doi.org/10.1007/s10596-007-9044-z · Zbl 1117.74016 [19] Phillips, P.; Wheeler, M., A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci., 12, 4, 417-435, (2008), URL http://dx.doi.org/10.1007/s10596-008-9082-1 · Zbl 1155.74048 [20] 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), URL http://dx.doi.org/10.1137/15M1009822 · Zbl 1326.76054 [21] Showalter, R., Diffusion in poro-elastic media, J. Math. Anal. Appl., 251, 1, 310-340, (2000) · Zbl 0979.74018 [22] Lipnikov, K., Numerical methods for the Biot model in poroelasticity, (2002), University of Houston, (Ph.D. thesis) [23] Yi, S.-Y., A coupling of nonconforming and mixed finite element methods for biot’s consolidation model, Numer. Methods Partial Differential Equations, 29, 5, 1749-1777, (2013), URL http://dx.doi.org/10.1002/num.21775 · Zbl 1274.74455 [24] Di Pietro, D.; Lemaire, S., An extension of the couzeix-Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow, Math. Comp., 84, 291, 1-31, (2015) · Zbl 1308.74145 [25] Kuznetsov, Y.; Repin, S., New mixed finite element method on polygonal and polyhedral meshes, Russian J. Numer. Anal. Math. Modelling, 18, 3, 261-278, (2003), URL http://dx.doi.org/10.1163/156939803322380846 · Zbl 1048.65113 [26] Yi, S.-Y., Convergence analysis of a new mixed finite element method for biot’s consolidation model, Numer. Methods Partial Differential Equations, 1189-1210, (2014), URL http://dx.doi.org/10.1002/num.21865 · Zbl 1350.74024 [27] J.J. Lee, Robust error analysis of coupled mixed methods for biot’s consolidation model. ArXiv Preprint arXiv:1512.02038. · Zbl 1368.65234 [28] Crouzeix, M.; Raviart, P.-A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Français. Autom. Inform. Rech. Opér. Sér. Rouge, 7, R-3, 33-75, (1973) · Zbl 0302.65087 [29] Raviart, P.-A.; Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, (Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Lecture Notes in Math., vol. 606, (1977), Springer Berlin), 292-315 [30] Nédélec, J.-C., A new family of mixed finite elements in $$\mathbf{R}^3$$, Numer. Math., 50, 1, 57-81, (1986) · Zbl 0625.65107 [31] Nédélec, J.-C., Mixed finite elements in $$\mathbf{R}^3$$, Numer. Math., 35, 3, 315-341, (1980) · Zbl 0419.65069 [32] Falk, R. S., Nonconforming finite element methods for the equations of linear elasticity, Math. Comp., 57, 196, 529-550, (1991), URL http://dx.doi.org/10.2307/2938702 · Zbl 0747.73044 [33] Falk, R. S.; Morley, M. E., Equivalence of finite element methods for problems in elasticity, SIAM J. Numer. Anal., 27, 6, 1486-1505, (1990), URL http://dx.doi.org/10.1137/0727086 · Zbl 0722.73068 [34] Hansbo, P.; Larson, M. G., Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity, M2AN Math. Model. Numer. Anal., 37, 1, 63-72, (2003), URL http://dx.doi.org/10.1051/m2an:2003020 · Zbl 1137.65431 [35] Brenner, S. C., Korn’s inequalities for piecewise $$H^1$$ vector fields, Math. Comp., 73, 247, 1067-1087, (2004), URL http://dx.doi.org/10.1090/S0025-5718-03-01579-5 · Zbl 1055.65118 [36] Mardal, K.-A.; Winther, R., An observation on korn’s inequality for nonconforming finite element methods, Math. Comp., 75, 253, 1-6, (2006), URL http://dx.doi.org/10.1090/S0025-5718-05-01783-7 · Zbl 1086.65112 [37] Brezzi, F.; Fortin, M.; Marini, L. D., Error analysis of piecewise constant pressure approximations of darcy’s law, Comput. Methods Appl. Mech. Engrg., 195, 13-16, 1547-1559, (2006), URL http://dx.doi.org/10.1016/j.cma.2005.05.027 · Zbl 1116.76051 [38] Baranger, J.; Maitre, J.-F.; Oudin, F., Connection between finite volume and mixed finite element methods, RAIRO Modél. Math. Anal. Numér., 30, 4, 445-465, (1996) · Zbl 0857.65116 [39] Brandts, J.; Korotov, S.; Krízek, M.; Šolc, J., On nonobtuse simplicial partitions, SIAM Rev., 51, 2, 317-335, (2009) · Zbl 1172.51012 [40] Micheletti, S.; Sacco, R.; Saleri, F., On some mixed finite element methods with numerical integration, SIAM J. Sci. Comput., 23, 1, 245-270, (2001) · Zbl 0992.65126 [41] Wheeler, M.; Xue, G.; Yotov, I., A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra, Numer. Math., 121, 1, 165-204, (2012) · Zbl 1277.65100 [42] Thomée, V., (Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, vol. 25, (2006), Springer-Verlag Berlin) · Zbl 1105.65102
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