×

New stabilized discretizations for poroelasticity and the Stokes’ equations. (English) Zbl 1440.76027

Summary: In this work, we consider the popular P1-RT0-P0 discretization of the three-field formulation of Biot’s consolidation problem. Since this finite-element formulation is not uniformly stable with respect to the physical parameters, several issues arise in numerical simulations. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur. To alleviate such problems, we consider a well-known stabilization technique with face bubble functions. We then design a perturbation of the bilinear form, which allows for local elimination of the bubble functions. We further prove that such perturbation is consistent and the resulting scheme has optimal approximation properties for both Biot’s model as well as the Stokes’ equations. For the former, the number of degrees of freedom is the same as for the classical P1-RT0-P0 discretization and for the latter (Stokes’ equations) the number of degrees of freedom is the same as for a P1-P0 discretization. We present numerical tests confirming the theoretical results for the poroelastic and the Stokes’ test problems.

MSC:

76D07 Stokes and related (Oseen, etc.) flows
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Terzaghi, K., Theoretical Soil Mechanics (1943), Wiley: New York
[2] Biot, M. A., General Theory of three-dimensional 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] Showalter, R., Diffusion in poro-elastic media, J. Math. Anal. Appl., 251, 1, 310-340 (2000) · Zbl 0979.74018
[5] Ženíšek, A., The existence and uniqueness theorem in Biot’s consolidation theory, Apl. Mat., 29, 3, 194-211 (1984) · Zbl 0557.35005
[6] 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) · Zbl 1023.76032
[7] Gaspar, F. J.; Lisbona, F. J.; Vabishchevich, P. N., Staggered grid discretizations for the quasi-static Biot’s consolidation problem, Appl. Numer. Math., 56, 6, 888-898 (2006) · Zbl 1091.76047
[8] Nordbotten, J. M., Cell-centered finite volume discretizations for deformable porous media, Internat. J. Numer. Methods Engrg., 100, 6, 399-418 (2014) · Zbl 1352.76072
[9] Nordbotten, J. M., Stable cell-centered finite volume discretization for biot equations, SIAM J. Numer. Anal., 54, 2, 942-968 (2016) · Zbl 1382.76187
[10] Lewis, R.; Schrefler, B., The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media (1998), Wiley: New York · Zbl 0935.74004
[11] 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) · Zbl 0760.73068
[12] 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) · Zbl 0791.76047
[13] 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) · Zbl 0854.76053
[14] 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) · Zbl 1425.74164
[15] Hu, X.; 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) · Zbl 1381.76175
[16] 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) · Zbl 1398.65046
[17] Oyarzúa, R.; Ruiz-Baier, R., Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal., 54, 5, 2951-2973 (2016) · Zbl 1457.65210
[18] Lee, J. J.; Mardal, K.-A.; Winther, R., Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J. Sci. Comput., 39, 1, A1-A24 (2017) · Zbl 1381.76183
[19] Lee, J. J., Robust error analysis of coupled mixed methods for biot’s consolidation model, J. Sci. Comput., 69, 2, 610-632 (2016) · Zbl 1368.65234
[20] Mikelić, A.; Wheeler, M. F., Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17, 3, 455-461 (2013) · Zbl 1392.35235
[21] Both, J. W.; Borregales, M.; Nordbotten, J. M.; Kumar, K.; Radu, F. A., Robust fixed stress splitting for Biot’s equations in heterogeneous media, Appl. Math. Lett., 68, 101-108 (2017) · Zbl 1383.74025
[22] 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) · Zbl 1117.74015
[23] 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) · Zbl 1117.74016
[24] Phillips, P.; Wheeler, M., A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci., 12, 4, 417-435 (2008) · Zbl 1155.74048
[25] Castelleto, N.; White, J. A.; Ferronato, M., Scalable algorithms for three-field mixed finite element coupled poromechanics, J. Comput. Phys., 327, 894-918 (2016) · Zbl 1373.76312
[26] Bause, M.; Radu, F.; Kocher, U., Space-time finite element approximation of the Biot poroelasticity system with iterative coupling, Comput. Methods Appl. Mech. Engrg., 320, 745-768 (2017) · Zbl 1439.74389
[27] Almani, T.; Kumar, K.; Dogru, A.; Singh, G.; Wheeler, M., Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics, Comput. Methods Appl. Mech. Engrg., 311, 1, 180-207 (2016) · Zbl 1439.74183
[28] Lipnikov, K., Numerical methods for the Biot model in poroelasticity (2002), University of Houston, Ph.D. thesis
[29] Brezzi, F.; Fortin, M., (Mixed and hybrid finite element methods. Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol.15 (1991), Springer-Verlag, New York), x+350 · Zbl 0788.73002
[30] Boffi, D.; Brezzi, F.; Demkowicz, L. F.; Durán, R. G.; Falk, R. S.; Fortin, M., (Mixed finite elements, compatibility conditions, and applications. Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics, vol.1939 (2008), Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence), x+235, Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26-July 1, 2006, Edited by Boffi and Lucia Gastaldi
[31] Boffi, D.; Brezzi, F.; Fortin, M., (Mixed finite element methods and applications. Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44 (2013), Springer: Springer Heidelberg), xiv+685 · Zbl 1277.65092
[41] Brezzi, F.; Douglas, Jr., J.; Durán, R.; Fortin, M., Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51, 2, 237-250 (1987) · Zbl 0631.65107
[42] Monk, P., Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, xiv+450 (2003), Oxford University Press, New York · Zbl 1024.78009
[43] X. Hu, J.H. Adler, L.T. Zikatanov, HAZmath: A Simple Finite Element, Graph, and Solver Library, 2014-2017 https://bitbucket.org/XiaozheHu/hazmath/wiki/Home; X. Hu, J.H. Adler, L.T. Zikatanov, HAZmath: A Simple Finite Element, Graph, and Solver Library, 2014-2017 https://bitbucket.org/XiaozheHu/hazmath/wiki/Home
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.