Robust approximation of generalized Biot-Brinkman problems. (English) Zbl 1503.65231
Summary: The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical examples.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 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 |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
| 65F10 | Iterative numerical methods for linear systems |
| 76S05 | Flows in porous media; filtration; seepage |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74L10 | Soil and rock mechanics |
| 35A15 | Variational methods applied to PDEs |
| 35B25 | Singular perturbations in context of PDEs |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
poromechanics; finite element method; preconditioning; Biot equations; Brinkman approximation; multiple-network poroelasticityReferences:
| [1] | Arnold, DN; Brezzi, F.; Cockburn, B.; Marini, LD, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779 (2002) · Zbl 1008.65080 |
| [2] | Arnold, DN; Falk, RS; Winther, R., Multigrid in H(div) and H(curl), Numer. Math., 85, 197-217 (2000) · Zbl 0974.65113 |
| [3] | Aznaran, F., Kirby, R., Farrell, P.: Transformations for Piola-mapped elements, arXiv preprint arXiv:2110.13224, (2021) |
| [4] | Bai, M.; Elsworth, D.; Roegiers, J-C, Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs, Water Resour. Res., 29, 1621-1633 (1993) |
| [5] | Barenblatt, G.; Zheltov, G.; Kochina, I., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], J. Appl. Math. Mech., 24, 5, 1286-1303 (1960) · Zbl 0104.21702 |
| [6] | Barnafi, N.; Zunino, P.; Dedè, L.; Quarteroni, A., Mathematical analysis and numerical approximation of a general linearized poro-hyperelastic model, Comput. Math. Appl., 91, 202-228 (2021) · Zbl 1524.65512 |
| [7] | Biot, M., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164 (1941) · JFM 67.0837.01 |
| [8] | Biot, M., Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26, 182-185 (1955) · Zbl 0067.23603 |
| [9] | Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications (2013), Heidelberg: Springer, Heidelberg · Zbl 1277.65092 |
| [10] | Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), pp. 129-151 · Zbl 0338.90047 |
| [11] | Brinkman, HC, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Flow Turbul. Combust., 1, 27-34 (1949) · Zbl 0041.54204 |
| [12] | Burtschell, B.; Moireau, P.; Chapelle, D., Numerical analysis for an energy-stable total discretization of a poromechanics model with inf-sup stability, Acta Math. Appl. Sin., 35, 28-53 (2019) · Zbl 1428.65026 |
| [13] | Chabiniok, R.; Wang, VY; Hadjicharalambous, M.; Asner, L.; Lee, J.; Sermesant, M.; Kuhl, E.; Young, AA; Moireau, P.; Nash, MP, Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics, Interface focus, 6, 20150083 (2016) |
| [14] | Chapelle, D.; Moireau, P., General coupling of porous flows and hyperelastic formulations-from thermodynamics principles to energy balance and compatible time schemes, Europ. J. Mech. B Fluids, 46, 82-96 (2014) · Zbl 1297.76157 |
| [15] | Chen, S.; Hong, Q.; Xu, J.; Yang, K., Robust block preconditioners for poroelasticity, Comput. Methods Appl. Mech. Eng., 369 (2020) · Zbl 1506.74104 |
| [16] | Chou, D.; Vardakis, J.; Guo, L.; Tully, B.; Ventikos, Y., A fully dynamic multi-compartmental poroelastic system: application to aqueductal stenosis, J. Biomech., 49, 2306-2312 (2016) |
| [17] | Darcy, H.: Les fontaines publiques de la ville de Dijon: exposition et application..., Victor Dalmont, 1856 |
| [18] | Eliseussen, E., Rognes, M.E., Thompson, T.B.: A-posteriori error estimation and adaptivity for multiple-network poroelasticity, arXiv preprint arXiv:2111.13456, (2021) |
| [19] | Farrell, PE; Knepley, MG; Mitchell, L.; Wechsung, F., PCPATCH: software for the topological construction of multigrid relaxation methods, ACM Trans. Math. Softw., 47, 3, 1-22 (2021) · Zbl 07467956 |
| [20] | Guo, L.; Vardakis, J.; Lassila, T.; Mitolo, M.; Ravikumar, N.; Chou, D.; Lange, M.; Sarrami-Foroushani, A.; Tully, B.; Taylor, Z.; Varma, S.; Venneri, A.; Frangi, A.; Ventikos, Y., Subject-specific multi-poroelastic model for exploring the risk factors associated with the early stages of Alzheimer’s disease, Interface Focus, 8, 20170019 (2018) |
| [21] | Hansbo, P.; Larson, M., Discontinuous Galerkin and the Crouzeix-Raviart element application to elasticity, ESAIM Math. Modell. Numer. Anal., 37, 1, 63-72 (2003) · Zbl 1137.65431 |
| [22] | Hong, Q.; Kraus, J., Uniformly stable discontinuous Galerkin discretization and robust iterative solution methods for the Brinkman problem, SIAM J. Numer. Anal., 54, 2750-2774 (2016) · Zbl 1346.76068 |
| [23] | 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 |
| [24] | Hong, Q.; Kraus, J.; Lymbery, M.; Philo, F., Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelasticity models, Numer. Linear Algebra Appl., 26 (2019) · Zbl 1463.65372 |
| [25] | Hong, Q.; Kraus, J.; Lymbery, M.; Philo, F., Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biot’s consolidation and multiple-network poroelasticity models, Math. Models Methods Appl. Sci., 30, 2523-2555 (2020) · Zbl 1471.65143 |
| [26] | Hong, Q., Kraus, J., Lymbery, M., Philo, F.: A new framework for the stability analysis of perturbed saddle-point problems and applications, arXiv:2103.09357v3 [math.NA], (2021) · Zbl 1504.65203 |
| [27] | Hong, Q.; Kraus, J.; Lymbery, M.; Wheeler, MF, Parameter-robust convergence analysis of fixed-stress split iterative method for multiple-permeability poroelasticity systems, Multiscale Model. Simulat., 18, 916-941 (2020) · Zbl 1447.65077 |
| [28] | Hong, Q.; Kraus, J.; Xu, J.; Zikatanov, L., A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations, Numer. Math., 132, 23-49 (2016) · Zbl 1338.76054 |
| [29] | Hong, Q.; Wang, F.; Wu, S.; Xu, J., A unified study of continuous and discontinuous Galerkin methods, SCI. CHINA Math., 62, 1-32 (2019) · Zbl 1412.65214 |
| [30] | Howell, JS; Walkington, NJ, Inf-sup conditions for twofold saddle point problems, Numer. Math., 118, 663 (2011) · Zbl 1230.65128 |
| [31] | Kedarasetti, R., Drew, P.J., Costanzo, F.: Arterial vasodilation drives convective fluid flow in the brain: a poroelastic model, bioRxiv, (2021) |
| [32] | Khaled, M., Beskos, D., Aifantis, E.: On the theory of consolidation with double porosity. 3. a finite-element formulation, (1984), pp. 101-123 · Zbl 0586.73116 |
| [33] | Kraus, J.; Lederer, PL; Lymbery, M.; Schöberl, J., Uniformly well-posed hybridized discontinuous Galerkin/hybrid mixed discretizations for Biot’s consolidation model, Comput. Methods Appl. Mech. Engrg., 384 (2021) · Zbl 1507.76201 |
| [34] | Kumar, S.; Oyarzúa, R.; Ruiz-Baier, R.; Sandilya, R., Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity, ESAIM Math. Model. Numer. Anal., 54, 273-299 (2020) · Zbl 1511.65114 |
| [35] | Lee, J.; Mardal, K-A; Winther, R., Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J. Sci. Comput., 39, A1-A24 (2017) · Zbl 1381.76183 |
| [36] | Lee, JJ; Piersanti, E.; Mardal, K-A; Rognes, ME, A mixed finite element method for nearly incompressible multiple-network poroelasticity, SIAM J. Sci. Comput., 41, A722-A747 (2019) · Zbl 1417.65162 |
| [37] | Mardal, K-A; Rognes, ME; Thompson, TB, Accurate discretization of poroelasticity without Darcy stability, BIT Numer. Math., 61, 3, 941-976 (2021) · Zbl 1470.65169 |
| [38] | Nash, MP; Hunter, PJ, Computational mechanics of the heart, J. Elast. Phys. Sci. solids, 61, 113-141 (2000) · Zbl 1071.74659 |
| [39] | Piersanti, E.; Lee, JJ; Thompson, T.; Mardal, K-A; Rognes, ME, Parameter robust preconditioning by congruence for multiple-network poroelasticity, SIAM J. Sci. Comput., 43, B984-B1007 (2021) · Zbl 1486.65174 |
| [40] | Rajagopal, KR, On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. Models Methods Appl. Sci., 17, 215-252 (2007) · Zbl 1123.76066 |
| [41] | Rathgeber, F.; Ham, DA; Mitchell, L.; Lange, M.; Luporini, F.; Mcrae, ATT; Bercea, G-T; Markall, GR; Kelly, PHJ, Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Softw., 43, 3, 1-27 (2016) · Zbl 1396.65144 |
| [42] | Rodrigo, C.; Hu, X.; Ohm, P.; Adler, JH; Gaspar, FJ; Zikatanov, L., New stabilized discretizations for poroelasticity and the Stokes’ equations, Comput. Methods Appl. Mech. Eng., 341, 467-484 (2018) · Zbl 1440.76027 |
| [43] | Showalter, R., Poroelastic filtration coupled to Stokes flow, Lecture Notes in Pure and Appl, Math., 242, 229-241 (2010) · Zbl 1084.76070 |
| [44] | Støverud, KH; Alnæs, M.; Langtangen, HP; Haughton, V.; Mardal, K-A, 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, 686-698 (2016) |
| [45] | Tully, B.; Ventikos, Y., Cerebral water transport using multiple-network poroelastic theory: application to normal pressure hydrocephalus, J. Fluid Mech., 667, 188-215 (2011) · Zbl 1225.76317 |
| [46] | Vardakis, J.; Chou, D.; Tully, B.; Hung, C.; Lee, T.; Tsui, P.; Ventikos, Y., Investigating cerebral oedema using poroelasticity, Med. Eng. Phys., 38, 48-57 (2016) |
| [47] | Whitaker, S., Flow in porous media I: a theoretical derivation of Darcy’s law, Transp. Porous Med., 1, 3-25 (1986) |
| [48] | Wilson, R.; Aifantis, E., On the theory of consolidation with double porosity, Int. J. Eng. Sci., 20, 1009-1035 (1982) · Zbl 0493.76094 |
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.
