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Numerical analysis for an energy-stable total discretization of a poromechanics model with inf-sup stability. (English) Zbl 1428.65026

The authors derive a linearization of a general nonlinear poromechanical model consisting of a two-phase mixture in which a fluid phase and a solid phase coexist and interact at each point. The linearization is then studied in terms of the existence of solutions and an analysis of a full discretization based on employing finite elements in space and a backward Euler scheme in time. In particular, the obtained error estimate is uniform with respect to the compressibility parameter. The theoretical assumptions and results are illustrated by numerical experiments.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
76S05 Flows in porous media; filtration; seepage
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
93C20 Control/observation systems governed by partial differential equations

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References:

[1] Bensoussan, A., Delfour, M.C., Da Prato, G., Mitter, S.K. Representation and Control of Infinite Dimensional Systems. Birkhauser Verlag, second edition, 2007 · Zbl 1117.93002
[2] Brezzi, F., Fortin, M. Mixed and Hybrid Finite Element Methods. Springer, 1991 · Zbl 0788.73002
[3] Burman, E.; Fernández, M. A., Stabilized explicit coupling for fluid-structure interaction using Nitsche’s method, C.R. Math., 345, 467-472, (2007) · Zbl 1126.74047
[4] Burman, E.; Fernández, M. A., Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility, Comput. Methods Appl. Mech. Eng., 198, 766-784, (2009) · Zbl 1229.76045
[5] Burtschell, B.; Chapelle, D.; Moireau, P., Effective and energy-preserving time discretization for a general nonlinear poromechanical formulation, Comput. Struct., 182, 313-324, (2017)
[6] Chapelle, D.; Moireau, P., General coupling of porous flows and hyperelastic formulations-from thermodynamics principles to energy balance and compatible time schemes, Eur. J. Mech. B. Fluids, 46, 82-96, (2014) · Zbl 1297.76157
[7] Ciarlet, P.G. Mathematical Elasticity-Volume I: Three-Dimensional Elasticity. North-Holland, 1988 · Zbl 0648.73014
[8] Clément, P., Approximation by finite element functions using local regularization, Rev. Fr. Automat. Infor., 9, 77-84, (1975) · Zbl 0368.65008
[9] Costanzo, F.; Miller, S. T., An arbitrary Lagrangian-Eulerian finite element formulation for a poroelasticity problem stemming from mixture theory, Comput. Methods Appl. M., 323, 64-97, (2017)
[10] Coussy, O. Poromechanics. John Wiley and Sons, 2004 · Zbl 1120.74447
[11] Fernández, M. A.; Gerbeau, J. F.; Grandmont, C., A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid, Int. J. Numer. Methods Eng., 69, 794-821, (2007) · Zbl 1194.74393
[12] Gerbeau, J. F.; Vidrascu, M., A Quasi-Newton Algorithm Based on a Reduced Model for Fluid-Structure Interaction Problems in Blood Flows, ESAIM: M2AN, 37, 631-647, (2003) · Zbl 1070.74047
[13] Hecht, F., New development in FreeFem++, J. Numer. Math., 20, 251-265, (2012) · Zbl 1266.68090
[14] Tallec, P.; Hauret, P.; Pironneau, O. (ed.); Kuznetsov, Y. (ed.); Neittanmaki, P. (ed.), Energy conservation in fluid structure interactions, (2003)
[15] Lions, J.L., Magenes, E. Non-Homogeneous Boundary Value Problems and Applications, volume 1. Springer-Verlag, 1972 · Zbl 0227.35001
[16] Nitsche, J., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, 36, 9-15, (1971) · Zbl 0229.65079
[17] Vuong, A. T.; Ager, C.; Wall, W. A., Two finite element approaches for Darcy and Darcy-Brinkman flow through deformable porous media-Mixed method vs, NURBS based (isogeometric) continuity. Comput. Methods Appl. M., 305, 634-657, (2016)
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