We introduce a numerical method for the approximation of linear poroelasticity equations, representing the interaction between the non-viscous filtration flow of a fluid and the linear mechanical response of a porous medium. In the proposed formulation, the primary variables in the system are the solid displacement, the fluid pressure, the fluid flux, and the total pressure. A discontinuous finite volume method is designed for the approximation of solid displacement using a dual mesh, whereas a mixed approach is employed to approximate fluid flux and the two pressures. We focus on the stationary case and the resulting discrete problem exhibits a double saddle-point structure. Its solvability and stability are established in terms of bounds (and of norms) that do not depend on the modulus of dilation of the solid. We derive optimal error estimates in suitable norms, for all field variables; and we exemplify the convergence and locking-free properties of this scheme through a series of numerical tests.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2019063
Mots-clés : Biot problem, discontinuous finite volume methods, mixed finite elements, locking-free approximations, conservative schemes, error estimates
@article{M2AN_2020__54_1_273_0,
author = {Kumar, Sarvesh and Oyarz\'ua, Ricardo and Ruiz-Baier, Ricardo and Sandilya, Ruchi},
title = {Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {273--299},
publisher = {EDP-Sciences},
volume = {54},
number = {1},
year = {2020},
doi = {10.1051/m2an/2019063},
mrnumber = {4058208},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2019063/}
}
TY - JOUR AU - Kumar, Sarvesh AU - Oyarzúa, Ricardo AU - Ruiz-Baier, Ricardo AU - Sandilya, Ruchi TI - Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 273 EP - 299 VL - 54 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019063/ DO - 10.1051/m2an/2019063 LA - en ID - M2AN_2020__54_1_273_0 ER -
%0 Journal Article %A Kumar, Sarvesh %A Oyarzúa, Ricardo %A Ruiz-Baier, Ricardo %A Sandilya, Ruchi %T Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 273-299 %V 54 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019063/ %R 10.1051/m2an/2019063 %G en %F M2AN_2020__54_1_273_0
Kumar, Sarvesh; Oyarzúa, Ricardo; Ruiz-Baier, Ricardo; Sandilya, Ruchi. Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 1, pp. 273-299. doi : 10.1051/m2an/2019063. http://www.numdam.org/articles/10.1051/m2an/2019063/
[1] , and , Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation. Int. J. Numer. Methods Eng. 75 (2008) 1282–1300. | DOI | MR
[2] , and , 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. | DOI | MR
[3] , , and , A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model. Numer. Math. 140 (2018) 513–553. | DOI | MR
[4] , , , and , Rotation-based mixed formulations for an elasticity-poroelasticity interface problem. Submitted preprint (2019). Available from https://www.ci2ma.udec.cl/publicaciones/prepublicaciones/. | MR
[5] , , and , Mixed displacement-rotation-pressure formulations for linear elasticity. Comput. Methods Appl. Mech. Eng. 344 (2019) 71–94. | DOI | MR
[6] , An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. | DOI | MR | Zbl
[7] , , and , Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. | DOI | MR | Zbl
[8] , and , Finite volume coupling strategies for the solution of a Biot consolidation model. Comput. Geotech. 55 (2014) 494–505. | DOI
[9] , , and , Weakly imposed symmetry and robust preconditioners for Biot’s consolidation model. Comput. Methods Appl. Math. 17 (2017) 377–396. | DOI | MR
[10] , , and , Stabilized lowest-order finite element approximation for linear three-field poroelasticity. SIAM J. Sci. Comput. 37 (2015) A2222–A2245. | DOI | MR
[11] , , and , A stabilized finite element method for finite-strain three-field poroelasticity. Comput. Mech. 60 (2017) 51–68. | DOI | MR
[12] , Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26 (1955) 182–185. | DOI | MR | Zbl
[13] , and , A nonconforming high-order method for the Biot problem on general meshes. SIAM J. Sci. Comput. 38 (2016) A1508–A1537. | DOI | MR
[14] , and , Mixed finite element methods and applications. In: Vol. 44 of Springer Series in Computational Mathematics. Springer (2010). | MR | Zbl
[15] , Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Verlag (2010). | MR | Zbl
[16] , and , Dimensional model reduction for flow through fractures in poroelastic media. ESAIM: M2AN 51 (2017) 1429–1471. | Numdam | MR
[17] , and , Discontinuous finite volume element discretization for coupled flow-transport problems arising in models of sedimentation. J. Comput. Phys. 299 (2015) 446–471. | DOI | MR
[18] , and , Comparison results and unified analysis for first-order finite volume element methods for a Poisson model problem. IMA J. Numer. Anal. 36 (2016) 1120–1142. | DOI | MR
[19] , and , Scalable algorithms for three-field mixed finite element coupled poromechanics. J. Comput. Phys. 327 (2016) 894–918. | DOI | MR
[20] and , General coupling of porous flows and hyperelastic formulations – From thermodynamics principles to energy balance and compatible time schemes. Eur. J. Mech. B/Fluids 46 (2014) 82–96. | DOI | MR | Zbl
[21] , and , Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem. Appl. Math. Comput. 219 (2013) 9043–9056. | MR | Zbl
[22] , and , A second-order hybrid finite volume method for solving the Stokes equation. Appl. Numer. Math. 119 (2017) 213–224. | DOI | MR
[23] , and , A second-order finite-volume-element scheme for linear poroelasticity. Submitted preprint (2019). Available from https://www.ci2ma.udec.cl/publicaciones/prepublicaciones/.
[24] , , and , A locally conservative stabilized continuous Galerkin finite element method for two-phase flow in poroelastic subsurfaces. J. Comput. Phys. 347 (2017) 78–98. | DOI | MR
[25] and , Mathematical aspects of discontinuous Galerkin methods. In: Vol. 69 of Mathématiques & Applications. Springer, Heidelberg (2012). | DOI | MR | Zbl
[26] and , Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units. Opt. Express 17 (2009) 20178–20190. | DOI
[27] , and , Analysis of a multiphysics finite element method for a poroelasticity model. IMA J. Numer. Anal. 38 (2018) 330–359. | DOI | MR
[28] , A high-order HDG method for the Biot’s consolidation model. Comput. Math. Appl. 77 (2019) 237–252. | DOI | MR
[29] , A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014). | MR | Zbl
[30] and , Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11) (2009) 1309–1331. | DOI | MR | Zbl
[31] and , Finite element approximation of the Navier–Stokes equations. In: Vol. 749 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York (1979). | DOI | MR | Zbl
[32] , , and , A lubrication fracture model in a poro-elastic medium. Math. Models Methods Appl. Sci. 25 (2015) 587–645. | DOI | MR
[33] and , Parameter-robust stability of classical three-field formulation of Biot’s consolidation model. Electron. Trans. Numer. Anal. 48 (2018) 202–226. | DOI | MR
[34] , , and , Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelasticity models. Numer. Linear Alg. Appl. 26 (2019) e2242. | DOI | MR
[35] , , and , A non-conforming finite element method for the Biot’s consolidation model in poroelasticity. J. Comput. Appl. Math. 310 (2017) 143–154. | DOI | MR
[36] and , A finite element method with strong mass conservation for Biot’s linear consolidation model. J. Sci. Comput. 77 (2018) 1762–1779. | DOI | MR
[37] and , Equal order discontinuous finite volume element methods for the Stokes problem. J. Sci. Comput. 65 (2015) 956–978. | DOI | MR
[38] and , Stabilized discontinuous finite element approximations for Stokes equations. J. Comput. Appl. Math. 198 (2007) 236–252. | DOI | MR | Zbl
[39] , and , Parameter-robust discretization and preconditioning of Biot’s consolidation model. SIAM J. Sci. Comput. 39 (2017) A1–A24. | DOI | MR
[40] , , and , A mixed finite element method for nearly incompressible multiple-network poroelasticity. SIAM J. Sci. Comput. 41 (2019) A722–A747. | DOI | MR
[41] and , The Finite Element Method in the Deformation of and Consolidation of Porous Media. Wiley & Sons, Chichester (1987). | Zbl
[42] , , and , On a coupled discontinuous/continuous Galerkin framework and an adaptive penalty scheme for poroelasticity problems. Comput. Methods Appl. Mech. Eng. 198 (2009) 3499–3510. | DOI | Zbl
[43] and , Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18 (2011) 1–40. | DOI | MR | Zbl
[44] , and , Asymptotic behavior of semi discrete finite-element approximations of Biot’s consolidation problem. SIAM J. Numer. Anal. 33 (1996) 1065–1083. | DOI | MR | Zbl
[45] , On finite volume discretization of the three-dimensional Biot poroelasticity system in multilayer domains. Comput. Methods Appl. Math. 6 (2006) 306–325. | DOI | MR | Zbl
[46] and , On a multigrid solver for the three-dimensional Biot poroelasticity system in multilayered domains. Comput. Vis. Sci. 11 (2008) 77–87. | DOI | MR
[47] and , Locking-free finite element methods for poroelasticity. SIAM J. Numer. Anal. 54 (2016) 2951–2973. | DOI | MR
[48] , and , Effective governing equations for poroelastic growing media. Q. J. Mech. Appl. Math. 67 (2014) 69–91. | DOI | MR
[49] and , Solute transport through a deforming porous medium. Int. J. Numer. Anal. Methods Geomech. 26 (2002) 683–717. | DOI | Zbl
[50] and , A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity. Comput. Geosci. 12 (2008) 417–435. | DOI | MR | Zbl
[51] , and , An active poroelastic model for mechanochemical patterns in protoplasmic droplets of physarum polycephalum.PLoS One 9 (2014) e99220. | DOI
[52] , , , and , Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis. Comput. Math. Appl. 73 (2017) 1593–1610. | DOI | MR | Zbl
[53] , and , Error analysis of primal discontinuous Galerkin methods for a mixed formulation of the Biot equations. Comput. Math. Appl. 73 (2017) 666–683. | DOI | MR | Zbl
[54] , , and , Stability and monotonicity for some discretizations of the Biot’s consolidation model. Comput. Methods Appl. Mech. Eng. 298 (2016) 183–204. | DOI | MR | Zbl
[55] and , Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models. J. Comput. Phys. 322 (2016) 666–688. | DOI | MR | Zbl
[56] , , and , A poroelastic mixture model of mechanobiological processes in biomass growth: theory and application to tissue engineering. Meccanica 52 (2017) 3273–3297. | DOI | MR | Zbl
[57] , Diffusion in poro-elastic media. J. Math. Anal. Appl. 251 (2000) 310–340. | DOI | MR | Zbl
[58] , , , and , Poro-elastic modeling of Syringomyelia – a systematic study of the effects of pia mater, central canal, median fissure, white and gray matter on pressure wave propagation and fluid movement within the cervical spinal cord. Comput. Methods Biomech. Biomed. Eng. 19 (2016) 686–698. | DOI
[59] and , A coupling of weak Galerkin and mixed finite element methods for poroelasticity. Comput. Math. Appl. 73 (2017) 804–823. | DOI | MR | Zbl
[60] , , , , , and , Investigating cerebral oedema using poroelasticity. Med. Eng. Phys. 38 (2016) 48–57. | DOI
[61] , and , A general approach for modeling interacting flow through porous media under finite deformations. Comput. Methods Appl. Mech. Eng. 283 (2015) 1240–1259. | DOI | MR | Zbl
[62] , and , Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity. Comput. Geosci. 18 (2014) 57–75. | DOI | MR | Zbl
[63] and , 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. | DOI | Zbl
[64] , A discontinuous finite volume method for the Stokes problem. SIAM J. Numer. Anal. 44 (2006) 183–198. | DOI | MR | Zbl
[65] , A Coupling of nonconforming and mixed finite element methods for Biot’s consolidation model. Numer. Methods Part. Diff. Equ. 29 (2013) 1749–1777. | DOI | MR | Zbl
[66] , Convergence analysis of a new mixed finite element method for Biot’s consolidation model. Numer. Methods Part. Diff. Equ. 30 (2014) 1189–1210. | DOI | MR | Zbl
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