## A high-order HDG method for the Biot’s consolidation model.(English)Zbl 1442.65257

Summary: We propose a novel high-order HDG method for the Biot’s consolidation model in poroelasticity. We present optimal $$h$$-version error analysis for both the semi-discrete and full-discrete (combined with temporal backward differentiation formula) schemes. Numerical tests are provided to demonstrate the performance of the method.

### MSC:

 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 76S05 Flows in porous media; filtration; seepage 35Q35 PDEs in connection with fluid mechanics

### Keywords:

HDG; divergence-conforming; fully discrete; poroelasticity
Full Text:

### References:

 [1] Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164 (1941) · JFM 67.0837.01 [2] Biot, M. A., Theory of deformation of a porous viscoelastic anisotropic solid, J. Appl. Phys., 27, 459-467 (1956) [3] Biot, M. A., Theory of finite deformations of pourous solids, Indiana Univ. Math. J., 21, 597-620 (1971) · Zbl 0229.76065 [4] Zheng, Y.; Burridge, R.; Rurns, D., Reservoir Simulation with the Finite Element Method using Biot Poroelastic Approach (2003), Massachusetts Institute of Technology. Earth Resources Laboratory [5] Strehlow, K.; Gottsmann, J.-H.; Rust, A.-C., Poroelastic responses of confined aquifers to subsurface strain and their use for volcano monitoring, Solid Earth, 6, 1207-1229 (2015) [6] Young, J.; Rivière, B.; Cox Jr, C. S.; Uray, K., A mathematical model of intestinal oedema formation, Math. Med. Biol., 31, 1-15 (2014) · Zbl 1304.92073 [7] Phillips, P. J.; Wheeler, M. F., A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. I. The continuous in time case, Comput. Geosci., 11, 131-144 (2007) · Zbl 1117.74015 [8] Murad, M. A.; Loula, A. F.D., Improved accuracy in finite element analysis of Biot’s consolidation problem, Comput. Methods Appl. Mech. Engrg., 95, 359-382 (1992) · Zbl 0760.73068 [9] Murad, M. A.; Loula, A. F.D., On stability and convergence of finite element approximations of Biot’s consolidation problem, Int. J. Numer. Methods Eng., 37, 645-667 (1994) · Zbl 0791.76047 [10] 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, 1065-1083 (1996) · Zbl 0854.76053 [11] Lewis, R. W.; Schrefler, B. A., (The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, Wiley Ser. Number. Methods Engrg., vol. 37 (1998), John Wiley: John Wiley New York) · Zbl 0935.74004 [12] Phillips, P. J.; Wheeler, M. F., A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. II. The discrete-in-time case, Comput. Geosci., 11, 145-158 (2007) · Zbl 1117.74016 [13] Phillips, P. J.; Wheeler, M. F., A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci., 12, 417-435 (2008) · Zbl 1155.74048 [14] Liu, R.; Wheeler, M. F.; Dawson, C. N.; Dean, R. H., On a coupled discontinous/continuous Galerkin framework and an adaptive penalty scheme for poroelasticity problems, Comput. Methods Appl. Mech. Engrg., 198, 3499-3510 (2009) · Zbl 1230.74189 [15] Chen, Y.; Luo, Y.; Feng, M., Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem, Appl. Math. Comput., 219, 9043-9056 (2013) · Zbl 1290.74038 [16] Wheeler, M.; Xue, G.; Yotov, I., Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity, Comput. Geosci., 18, 57-75 (2014) · Zbl 1395.65093 [17] Rivière, B.; Tan, J.; Thompson, T., Error analysis of primal discontinuous Galerkin methods for a mixed formulation of the Biot equations, Comput. Math. Appl., 73, 666-683 (2017) · Zbl 1368.65195 [18] Korsawe, J.; Starke, G., A least-squares mixed finite element method for Biot’s consolidation problem in porous media, SIAM J. Numer. Anal., 43, 318-339 (2005) · Zbl 1086.76041 [19] Wan, J., Stabilized Finite Element Method for Coupled Geomechanics and Multiphase Flow (2002), Stanford University: Stanford University Stanford, CA, (Ph.D. Thesis) [20] White, J. A.; Borja, R. I., Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients, Comput. Methods Appl. Mech. Engrg., 197, 4353-4366 (2008) · Zbl 1194.74480 [21] Berger, L.; Bordas, R.; Kay, D.; Tavener, S., Stabilized lowest-order finite element approximation for linear three-field poroelasticity, SIAM J. Sci. Comput., 37, A2222-A2245 (2015) · Zbl 1326.76054 [22] Berger, L.; Bordas, R.; Kay, D.; Tavener, S., A stabilized finite element method for finite-strain three-field poroelasticity, Comput. Mech., 60, 51-68 (2017) · Zbl 1386.74134 [23] Ferronato, M.; Castelletto, N.; Gambolati, G., A fully coupled 3-D mixed finite element model of Biot consolidation, J. Comput. Phys., 229, 4813-4830 (2010) · Zbl 1305.76055 [24] Yi, S.-Y., Convergence analysis of a new mixed finite element method for Biot’s consolidation model, Numer. Methods Partial Differential Equations, 30, 1189-1210 (2014) · Zbl 1350.74024 [25] Lee, J. J., Robust error analysis of coupled mixed methods for Biot’s consolidation model, J. Sci. Comput., 69, 610-632 (2016) · Zbl 1368.65234 [26] Oyarzúa, R.; Ruiz-Baier, R., Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal., 54, 2951-2973 (2016) · Zbl 1457.65210 [27] Baerland, T.; Lee, J. J.; Mardal, K.-A.; Winther, R., Weakly imposed symmetry and robust preconditioners for Biot’s consolidation model, Comput. Methods Appl. Math., 17, 377-396 (2017) · Zbl 1421.74095 [28] Lee, J. 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 [29] Yi, N.; Huang, Y.; Liu, H., A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation: energy conservation and boundary effect, J. Comput. Phys., 242, 351-366 (2013) · Zbl 1297.65122 [30] Boffi, D.; Botti, M.; Di Pietro, D. A., A nonconforming high-order method for the Biot problem on general meshes, SIAM J. Sci. Comput., 38, A1508-A1537 (2016) · Zbl 1337.76042 [31] 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 [32] Lehrenfeld, C., Hybrid Discontinuous Galerkin Methods for Solving Incompressible Flow Problems (2010), MathCCES/IGPM, RWTH Aachen, (Diploma Thesis) [33] Oikawa, I., A hybridized discontinuous Galerkin method with reduced stabilization, J. Sci. Comput., 65, 327-340 (2015) · Zbl 1331.65162 [34] 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) [35] Fu, G.; Lehrenfeld, C., A strongly conservative hybrid DG/Mixed FEM for the coupling of Stokes and Darcy flow, J. Sci. Comput. (2018) · Zbl 1406.65112 [37] Hairer, E.; Wanner, G., (Solving Ordinary Differential Equations. II. Solving Ordinary Differential Equations. II, Springer Series in Computational Mathematics, vol. 14 (2010), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1192.65097 [38] Phillips, P. J.; Wheeler, M. F., Overcoming the problem of locking in linear elasticity and poroleasticity: an heuristic approach, Comput. Geosci., 13, 5-12 (2009) · Zbl 1172.74017 [39] Lederer, P. L.; Lehrenfeld, C.; Schöberl, J., Hybrid discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part I, SIAM J. Numer. Anal, 56, 2070-2094 (2018) · Zbl 1402.35209 [40] Ainsworth, M.; Fu, G., Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations, J. Sci. Comput. (2018) · Zbl 1407.65275 [41] Wheeler, M. F., An $$H^{- 1}$$ Galerkin method for parabolic problems in a single space variable, SIAM J. Numer. Anal., 12, 803-817 (1975) · Zbl 0331.65075 [42] 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) · Zbl 1277.65092 [43] Chabaud, B.; Cockburn, B., Uniform-in-time superconvergence of HDG methods for the heat equation, Math. Comp., 81, 107-129 (2012) · Zbl 1251.65138 [44] Di Pietro, D. A.; Ern, A., Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations, Math. Comp., 79, 1303-1330 (2010) · Zbl 1369.76024 [45] Nevanlinna, O.; Odeh, F., Multiplier techniques for linear multistep methods, Numer. Funct. Anal. Optim., 3, 377-423 (1981) · Zbl 0469.65051 [46] Akrivis, G., Stability of implicit-explicit backward difference formulas for nonlinear parabolic equations, SIAM J. Numer. Anal., 53, 464-484 (2015) · Zbl 1312.65115 [47] Heywood, J. G.; Rannacher, R., Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27, 353-384 (1990) · Zbl 0694.76014 [48] Lubich, C.; Mansour, D.; Venkataraman, C., Backward difference time discretization of parabolic differential equations on evolving surfaces, IMA J. Numer. Anal., 33, 1365-1385 (2013) · Zbl 1401.65108 [49] Nguyen, N. C.; Peraire, J.; Cockburn, B., High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics, J. Comput. Phys., 230, 3695-3718 (2011) · Zbl 1364.76093 [50] Jaust, A.; Schütz, J., A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows, Comput. Fluids, 98, 177-185 (2014) · Zbl 1390.65109 [51] Carpenter, M. H.; Gottlieb, D.; Abarbanel, S.; Don, W. S., The theoretical accuracy of Runge-Kutta time discretizations for the initial – boundary value problem: a study of the boundary error, SIAM J. Sci. Comput., 16, 1241-1252 (1995) · Zbl 0839.65098 [54] Yi, S.-Y., A study of two modes of locking in poroelasticity, SIAM J. Numer. Anal., 55, 1915-1936 (2017) · Zbl 1430.74140 [55] Barry, S. I.; Mercer, G. N., Exact solutions for two-dimensional time-dependent flow and deformation within a poroelastic medium, Trans. ASME J. Appl. Mech., 66, 536-540 (1999) [56] Phillips, P. J., Finite Element Methods in Linear Poroelasticity: Theoretical and Computational Results (2005), The University of Texas at Austin: The University of Texas at Austin ProQuest LLC, Ann Arbor, MI, (Ph.D. Thesis) [57] Rodrigo, C.; Gaspar, F. J.; Hu, X.; Zikatanov, L. T., Stability and monotonicity for some discretizations of the Biot’s consolidation model, Comput. Methods Appl. Mech. Engrg., 298, 183-204 (2016) · Zbl 1425.74164
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.