## Uniformly well-posed hybridized discontinuous Galerkin/hybrid mixed discretizations for Biot’s consolidation model.(English)Zbl 07415276

Summary: We consider the quasi-static Biot’s consolidation model in a three-field formulation with the three unknown physical quantities of interest being the displacement $$\boldsymbol{u}$$ of the solid matrix, the seepage velocity $$\boldsymbol{v}$$ of the fluid and the pore pressure $$p$$. As conservation of fluid mass is a leading physical principle in poromechanics, we preserve this property using an $$\boldsymbol{H}(\operatorname{div})$$-conforming ansatz for $$\boldsymbol{u}$$ and $$\boldsymbol{v}$$ together with an appropriate pressure space. This results in Stokes and Darcy stability and exact, that is, pointwise mass conservation of the discrete model.
The proposed discretization technique combines a hybridized discontinuous Galerkin method for the elasticity subproblem with a mixed method for the flow subproblem, also handled by hybridization. The latter allows for a static condensation step to eliminate the seepage velocity from the system while preserving mass conservation. The system to be solved finally only contains degrees of freedom related to $$\boldsymbol{u}$$ and $$p$$ resulting from the hybridization process and thus provides, especially for higher-order approximations, a very cost-efficient family of physics-oriented space discretizations for poroelasticity problems. We present the construction of the discrete model, theoretical results related to its uniform well-posedness along with optimal error estimates and parameter-robust preconditioners as a key tool for developing uniformly convergent iterative solvers. Finally, the cost-efficiency of the proposed approach is illustrated in a series of numerical tests for three-dimensional test cases.

### MSC:

 76-XX Fluid mechanics 65-XX Numerical analysis

### Software:

NGSolve; Netgen; poroelasticity
Full Text:

### References:

 [1] Biot, M., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 2, 155-164 (1941) · JFM 67.0837.01 [2] Biot, M., Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26, 2, 182-185 (1955) · Zbl 0067.23603 [3] Sebaa, N.; Fellah, Z.; Fellah, M.; Ogam, E.; Mitri, F.; Depollier, C.; Laurikis, W., Application of the Biot model to ultrasound in bone: Inverse problem, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 55, 7, 1516-1523 (2008) [4] 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, Interf. Focus, 8, 1, Article 20170019 pp. (2018) [5] Murad, M.; Loula, A., Improved accuracy in finite element analysis of Biot’s consolidation problem, Comput. Methods Appl. Mech. Engrg., 95, 359-382 (1992) · Zbl 0760.73068 [6] Murad, M.; Loula, A., On stability and convergence of finite element approximations of Biot’s consolidation problem, Internat. J. Numer. Methods Engrg., 37, 645-667 (1994) · Zbl 0791.76047 [7] Rodrigo, C.; Hu, X.; Ohm, P.; Adler, J.; Gaspar, F.; Zikatanov, L., New stabilized discretizations for poroelasticity and the Stokes’ equations, Comput. Methods Appl. Mech. Engrg., 341, 467-484 (2018) · Zbl 1440.76027 [8] Nordbotten, J., Stable cell-centered finite volume discretization for Biot equations, SIAM J. Numer. Anal., 54, 942-968 (2016) · Zbl 1382.76187 [9] 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 [10] Oyarzúa, R.; Ruiz-Baier, R., Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal., 54, 2951-2973 (2016) · Zbl 1457.65210 [11] 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, 1, 273-299 (2020) · Zbl 07201584 [12] Girault, V.; Lu, X.; Wheeler, M., A posteriori error estimates for Biot system using Enriched Galerkin for flow, Comput. Methods Appl. Mech. Engrg., 369, Article 113185 pp. (2020) [13] Lee, S.; Wheeler, M., Enriched Galerkin methods for two-phase flow in porous media with capillary pressure, J. Comput. Phys., 367, 65-86 (2018) · Zbl 1415.76456 [14] Bause, M.; Radu, F.; Köcher, 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 [15] Radu, F.; Kumar, K.; Nordbotten, J.; Pop, I., A robust, mass conservative scheme for two-phase flow in porous media including Hölder continuous nonlinearities, IMA J. Numer. Anal., 38, 884-920 (2018) · Zbl 1477.76055 [16] Lee, J.; Piersanti, E.; Mardal, K.-A.; Rognes, M., A mixed finite element method for nearly incompressible multiple-network poroelasticity, SIAM J. Sci. Comput., 41, A722-A747 (2019) · Zbl 1417.65162 [17] Hong, Q.; Kraus, J.; Lymbery, M.; Wheeler, M., Parameter-robust convergence analysis of fixed-stress split iterative method for multiple-permeability poroelasticity systems, Multiscale Model. Simul., 18, 2, 916-941 (2020) · Zbl 1447.65077 [18] 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 [19] 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 [20] 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 [21] Yi, S.-Y., A coupling of nonconforming and mixed finite element methods for Biot’s consolidation model, Numer. Methods Partial Differ. Equ., 29, 5, 1749-1777 (2013) · Zbl 1274.74455 [22] Hu, X.; Rodrigo, C.; Gaspar, F.; Zikatanov, L., A nonconforming finite element method for the Biot’s consolidation model in poroelasticity, J. Comput. Appl. Math., 310, 143-154 (2017) · Zbl 1381.76175 [23] Hong, Q.; Kraus, J., Parameter-robust stability of classical three-field formulation of Biot’s consolidation model, ETNA - Electron. Trans. Numer. Anal., 48, 202-226 (2018) · Zbl 1398.65046 [24] Kanschat, G.; Riviere, B., A finite element method with strong mass conservation for Biot’s linear consolidation model, J. Sci. Comput., 77, 1762-1779 (2018) · Zbl 1407.65192 [25] Cockburn, B.; Kanschat, G.; Schotzau, D., A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp., 74, 251, 1067-1095 (2005) · Zbl 1069.76029 [26] Cockburn, B.; Kanschat, G.; Schötzau, D.; Schwab, C., Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal., 40, 1, 319-343 (2002) · Zbl 1032.65127 [27] Cockburn, B.; Kanschat, G.; Schötzau, D., A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput., 31, 1-2, 61-73 (2007) · Zbl 1151.76527 [28] Könnö, J.; Stenberg, R., Numerical computations with H(div)-finite elements for the Brinkman problem, Comput. Geosci., 16, 1, 139-158 (2012) · Zbl 1348.76100 [29] Könnö, J.; Stenberg, R., $$H ( \operatorname{div} )$$-conforming finite elements for the brinkman problem, Math. Models Methods Appl. Sci., 21, 11, 2227-2248 (2011) · Zbl 1331.76115 [30] Fu, G., A high-order HDG method for the Biot’s consolidation model, Comput. Math. Appl., 77, 1, 237-252 (2019) · Zbl 1442.65257 [31] Niu, C.; Rui, H.; Hu, X., A stabilized hybrid mixed finite element method for poroelasticity, Comput. Geosci. (2020) [32] Arnold, D. N.; Brezzi, F., Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér., 19, 1, 7-32 (1985) · Zbl 0567.65078 [33] Adler, J. H.; Gaspar, F. J.; Hu, X.; Rodrigo, C.; Zikatanov, L. T., Robust block preconditioners for Biot’s model, (Domain Decomposition Methods in Science and Engineering XXIV (2018), Springer International Publishing: Springer International Publishing Cham), 3-16 · Zbl 1442.65342 [34] Borregales, M.; Radu, F. A.; Kumar, K.; Nordbotten, J. M., Robust iterative schemes for non-linear poromechanics, Comput. Geosci., 22, 4, 1021-1038 (2018) · Zbl 1402.65109 [35] Rahrah, M.; Vermolen, F., A moving finite element framework for fast infiltration in nonlinear poroelastic media, Comput. Geosci., 25, 2, 793-804 (2021) · Zbl 1460.65126 [36] Borregales Reverón, M. A.; Kumar, K.; Nordbotten, J. M.; Radu, F. A., Iterative solvers for Biot model under small and large deformations, Comput. Geosci., 25, 2, 687-699 (2021) · Zbl 1460.65117 [37] Neunteufel, M.; Pechstein, A. S.; Schöberl, J., Three-field mixed finite element methods for nonlinear elasticity, Comput. Methods Appl. Mech. Engrg., 382, Article 113857 pp. (2021) [38] 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, 13, 2523-2555 (2020) · Zbl 1471.65143 [39] Terzaghi, K., Erdbaumechanik Auf Bodenphysikalischer Grundlage (1925), F. Deuticke · JFM 51.0655.07 [40] Boffi, D.; Brezzi, F.; Fortin, M., (Mixed Finite Element Methods and Applications. Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math., vol. 44 (2013), Springer, Heidelberg), xiv+685 · Zbl 1277.65092 [41] Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 2, 1319-1365 (2009) · Zbl 1205.65312 [42] Lehrenfeld, C.; Schöberl, J., High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows, Comput. Methods Appl. Mech. Engrg., 307, 339-361 (2016) · Zbl 1439.76081 [43] Lederer, P. L.; Lehrenfeld, C.; Schöberl, J., Hybrid discontinuous Galerkin methods with relaxed $$H ( \text{div} )$$-conformity for incompressible flows. Part II, ESAIM Math. Model. Numer. Anal., 53, 2, 503-522 (2019) · Zbl 1434.35058 [44] Lederer, P. L.; Lehrenfeld, C.; Schöberl, J., Hybrid discontinuous Galerkin methods with relaxed $$H ( \text{div} )$$-conformity for incompressible flows. Part I, SIAM J. Numer. Anal., 56, 4, 2070-2094 (2018) · Zbl 1402.35209 [45] 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., e2242 (2019) · Zbl 1463.65372 [46] Ženišek, A., The existence and uniqueness theorem in Biot’s consolidation theory, Apl. Mat., 29, 3, 194-211 (1984) · Zbl 0557.35005 [47] Ženišek, A., Finite element methods for coupled thermoelasticity and coupled consolidation of clay, RAIRO Anal. Numér., 18, 2, 183-205 (1984) · Zbl 0539.73005 [48] Showalter, R., Diffusion in poro-elastic media, J. Math. Anal. Appl., 251, 1, 310-340 (2000) · Zbl 0979.74018 [49] Hairer, E.; Lubich, C.; Roche, M., (The Numerical Solution of Differential-Algebraic Systems by Runge-KUtta Methods. The Numerical Solution of Differential-Algebraic Systems by Runge-KUtta Methods, Lecture Notes in Mathematics, vol. 1409 (1989), Springer-Verlag, Berlin), viii+139 · Zbl 0683.65050 [50] Babuška, I., Error-bounds for finite element method, Numer. Math., 16, 322-333 (1970) · Zbl 0214.42001 [51] 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 [52] Hong, Q.; Kraus, J., Uniformly stable discontinuous Galerkin discretization and robust iterative solution methods for the brinkman problem, SIAM J. Numer. Anal., 54, 5, 2750-2774 (2016) · Zbl 1346.76068 [53] Gopalakrishnan, J.; Lederer, P. L.; Schöberl, J., A mass conserving mixed stress formulation for the Stokes equations, IMA J. Numer. Anal., 40, 3, 1838-1874 (2019) · Zbl 1466.65189 [54] Gopalakrishnan, J.; Lederer, P. L.; Schöberl, J., A mass conserving mixed stress formulation for Stokes flow with weakly imposed stress symmetry, SIAM J. Numer. Anal., 58, 1, 706-732 (2020) · Zbl 1439.35368 [55] Fraeijs de Veubeke, B., (A Course in Elasticity. A Course in Elasticity, Applied Mathematical Sciences, vol. 29 (1979), Springer, Heidelberg), 330 · Zbl 0419.73001 [56] Mardal, K.-A.; Winther, R., Preconditioning discretizations of systems of partial differential equations, Numer. Linear Algebra Appl., 18, 1, 1-40 (2011) · Zbl 1249.65246 [57] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779 (2002) · Zbl 1008.65080 [58] Lehrenfeld, C., Hybrid discontinuous Galerkin methods for solving incompressible flow problems, Rheinisch-Westfal. Techn. Hochschule Aachen (2010) [59] Schöberl, J.; Lehrenfeld, C., Domain decomposition preconditioning for high order hybrid discontinuous Galerkin methods on tetrahedral meshes, (Apel, T.; Steinbach, O., Advanced Finite Element Methods and Applications (2013), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 27-56 · Zbl 1263.65120 [60] Schöberl, J., NETGEN an advancing front 2d/3D-mesh generator based on abstract rules, Comput. Vis. Sci., 1, 1, 41-52 (1997) · Zbl 0883.68130 [61] Schöberl, J., C++11 Implementation of Finite Elements in NGSolve (2014), Institute for Analysis and Scientific Computing, Vienna University of Technology [62] Cockburn, B.; Gopalakrishnan, J., A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal., 42, 1, 283-301 (2004) · Zbl 1084.65113
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.