Efficient solvers for hybridized three-field mixed finite element coupled poromechanics. (English) Zbl 1524.76451
Summary: We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement approximation strategy. Robustness, computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures.
MSC:
| 76S05 | Flows in porous media; filtration; seepage |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
References:
| [1] | Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164 · JFM 67.0837.01 |
| [2] | Coussy, O., Poromechanics (2004), Wiley: Wiley Chichester, UK |
| [3] | Girault, V.; Kumar, K.; Wheeler, M. F., Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium, Comput. Geosci., 20, 5, 997-1011 (2016) · Zbl 1391.76650 |
| [4] | Girault, V.; Wheeler, M. F.; Almani, T.; Dana, S., A priori error estimates for a discretized poro-elastic-elastic system solved by a fixed-stress algorithm, Oil Gas Sci. Technol. - Rev. IFP Energ. Nouv., 74, 24 (2019) |
| [5] | Showalter, R., Diffusion in poro-elastic media, J. Math. Anal. Appl., 251, 1, 310-340 (2000), URL http://www.sciencedirect.com/science/article/pii/S0022247X00970483 · Zbl 0979.74018 |
| [6] | Lipnikov, K., Numerical Methods for the Biot Model in Poroelasticity (2002), University of Houston, (PhD thesis) |
| [7] | Niu, C.; Rui, H.; Sun, M., A coupling of hybrid mixed and continuous Galerkin finite element methods for poroelasticity, Appl. Math. Comput., 347, 767-784 (2019) · Zbl 1428.74212 |
| [8] | Haga, J. B.; Osnes, H.; Langtangen, H. P., On the causes of pressure oscillations in low-permeable and low-compressible porous media, Int. J. Numer. Anal. Methods Geomech., 36, 12, 1507-1522 (2012) |
| [9] | Rodrigo, C.; Hu, X.; Ohm, P.; Adler, J. H.; Gaspar, F. J.; Zikatanov, L. T., New stabilized discretizations for poroelasticity and the Stokes’ equations, Comput. Methods Appl. Mech. Engrg., 341, 467-484 (2018) · Zbl 1440.76027 |
| [10] | Niu, C.; Rui, H.; Hu, X., A stabilized hybrid mixed finite element method for poroelasticity, Comput. Geosci. (2020) |
| [11] | Hermínio, H. T.; Maliska, C. R.; Ferronato, M.; Janna, C., A stabilized element-based finite volume method for poroelastic problems, J. Comput. Phys., 364, 49-72 (2018) · Zbl 1392.74090 |
| [12] | Camargo, J. T.; White, J. A.; Borja, R. I., A macroelement stabilization for mixed finite element/finite volume discretizations of multiphase poromechanics, Comput. Geosci. (2020) |
| [13] | Silvester, D. J.; Kechkar, N., Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem, Comput. Methods Appl. Mech. Engrg., 79, 1, 71-86 (1990) · Zbl 0706.76075 |
| [14] | Kuznetsov, Y.; Lipnikov, K.; Lyons, S.; Maliassov, S., Mathematical modeling and numerical algorithms for poroelastic problems, (Chen, Z.; Glowinski, R.; Li, K., Current Trends in Scientific Computing. Current Trends in Scientific Computing, Contemporary Mathematics, vol. 329 (2003), AMER MATHEMATICAL SOC: AMER MATHEMATICAL SOC P.O. BOX 6248, PROVIDENCE, RI 02940 USA), 191-202 · Zbl 1138.74393 |
| [15] | Gaspar, F. J.; Lisbona, F. J.; Oosterlee, C. W.; Wienands, R., A systematic comparison of coupled and distributive smoothing in multigrid for the poroelasticity system, Numer. Linear Algebr. Appl., 11, 2-3, 93-113 (2004) · Zbl 1164.65344 |
| [16] | Bergamaschi, L.; Ferronato, M.; Gambolati, G., Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations, Comput. Methods Appl. Mech. Engrg., 196, 25-28, 2647-2656 (2007) · Zbl 1173.76330 |
| [17] | Bergamaschi, L.; Ferronato, M.; Gambolati, G., Mixed constraint preconditioners for the iterative solution to FE coupled consolidation equations, J. Comput. Phys., 227, 9885-9897 (2008) · Zbl 1154.65015 |
| [18] | Ferronato, M.; Castelletto, N.; Gambolati, G., A fully coupled 3-D mixed finite element model of Biot consolidation, J. Comput. Phys., 229, 12, 4813-4830 (2010) · Zbl 1305.76055 |
| [19] | White, J. A.; Borja, R. I., Block-preconditioned Newton-Krylov solvers for fully coupled flow and geomechanics, Comput. Geosci., 15, 4, 647-659 (2011) · Zbl 1367.76034 |
| [20] | Axelsson, O.; Blaheta, R.; Byczanski, P., Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices, Comput. Vis. Sci., 15, 4, 191-207 (2012) · Zbl 1388.74035 |
| [21] | Turan, E.; Arbenz, P., Large scale micro finite element analysis of 3D bone poroelasticity, Parallel Comput., 40, 7, 239-250 (2014) |
| [22] | Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, C. W., Multigrid method for nonlinear poroelasticity equations, Comput. Vis. Sci., 17, 5, 255-265 (2015) · Zbl 1388.74013 |
| [23] | Castelletto, N.; White, J. A.; Ferronato, M., Scalable algorithms for three-field mixed finite element coupled poromechanics, J. Comput. Phys., 327, 894-918 (2016) · Zbl 1373.76312 |
| [24] | Gaspar, F. J.; Rodrigo, C., On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics, Comput. Methods Appl. Mech. Engrg., 326, 526-540 (2017) · Zbl 1439.74413 |
| [25] | Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, C. W., On an uzawa smoother in multigrid for poroelasticity equations, Numer. Linear Algebr. Appl., 24, 1, Article e2074 pp. (2017) · Zbl 1424.76041 |
| [26] | Lee, J. J.; Mardal, K.-A.; Winther, R., Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J. Sci. Comput., 39, 1, A1-A24 (2017) · Zbl 1381.76183 |
| [27] | Adler, J. H.; Gaspar, F. J.; Hu, X.; Rodrigo, C.; Zikatanov, L. T., Robust block preconditioners for biot’s model, (Bjøstad, P. E.; Brenner, S.; Halpern, L.; Kornhuber, R.; Kim, H. H.; Rahman, T.; Widlund, O. B., Domain Decomposition Methods in Science and Engineering XXIV. Domain Decomposition Methods in Science and Engineering XXIV, Lecture Notes in Computational Science and Engineering, vol. 125 (2018), Springer International Publishing), 3-16 · Zbl 1442.65342 |
| [28] | 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 |
| [29] | Ferronato, M.; Franceschini, A.; Janna, C.; Castelletto, N.; Tchelepi, H. A., A general preconditioning framework for coupled multi-physics problems with application to contact- and poro-mechanics, J. Comput. Phys., 398, Article 108887 pp. (2019) · Zbl 1453.65065 |
| [30] | Frigo, M.; Castelletto, N.; Ferronato, M., A relaxed physical factorization preconditioner for mixed finite element coupled poromechanics, SIAM J. Sci. Comput., 41, 4, B694-B720 (2019) · Zbl 1420.65032 |
| [31] | Adler, J. H.; Gaspar, F. J.; Hu, X.; Ohm, P.; Rodrigo, C.; Zikatanov, L. T., Robust preconditioners for a new stabilized discretization of the poroelastic equations, SIAM J. Sci. Comput., 42, 3, B761-B791 (2020) · Zbl 1448.65145 |
| [32] | Bui, Q. M.; Osei-Kuffuor, D.; Castelletto, N.; White, J. A., A scalable multigrid reduction framework for multiphase poromechanics of heterogeneous media, SIAM J. Sci. Comput., 42, 2, B379-B396 (2020) · Zbl 1435.65153 |
| [33] | Jha, B.; Juanes, R., A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics, Acta Geotech., 2, 3, 139-153 (2007) |
| [34] | Kim, J.; Tchelepi, H. A.; Juanes, R., Stability, accuracy and efficiency of sequential methods for coupled flow and geomechanics, SPE J., 16, 2, 249-262 (2011) · Zbl 1228.74106 |
| [35] | Kim, J.; Tchelepi, H. A.; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits, Comput. Methods Appl. Mech. Engrg., 200, 13, 1591-1606 (2011) · Zbl 1228.74101 |
| [36] | Mikelič, A.; Wheeler, M., Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17, 455-461 (2013) · Zbl 1392.35235 |
| [37] | Almani, T.; Kumar, K.; Dogru, A.; Singh, G.; Wheeler, M., Convergence analysis of multirate fixed-stress split interative schemes for coupling flow with geomechanics, Comput. Methods Appl. Mech. Engrg., 311, 180-207 (2016) · Zbl 1439.74183 |
| [38] | Both, J. W.; Borregales, M.; Nordbotten, J. M.; Kumar, K.; Radu, F. A., Robust fixed stress splitting for biot’s equations in heterogeneous media, Appl. Math. Lett., 68, 101-108 (2017) · Zbl 1383.74025 |
| [39] | 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 |
| [40] | Dana, S.; Ganis, B.; Wheeler, M. F., A multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs, J. Comput. Phys., 352, 1-22 (2018) · Zbl 1375.76085 |
| [41] | Dana, S.; Wheeler, M. F., Convergence analysis of two-grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media, Comput. Methods Appl. Mech. Engrg., 341, 788-806 (2018) · Zbl 1440.74122 |
| [42] | Hong, Q.; Kraus, J.; Lymbery, M.; Wheeler, M. F., 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 |
| [43] | Brezzi, F.; Bathe, K.-J., A discourse on the stability conditions for mixed finite element formulations, Comput. Methods Appl. Mech. Engrg., 82, 1-3, 27-57 (1990) · Zbl 0736.73062 |
| [44] | Elman, H.; Silvester, D. J.; Wathen, A., Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics (2014), Oxford University Press · Zbl 1304.76002 |
| [45] | Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 1-137 (2005) · Zbl 1115.65034 |
| [46] | Greenbaum, A.; Pták, V.; Strakoš, Z., Any nonincreasing convergence curve is possible for GMRES, SIAM J. Matrix Anal. Appl., 17, 465-469 (1996) · Zbl 0857.65029 |
| [47] | Chavent, G.; Jaffré, J., Mathematical Models and Finite Elements for Reservoir Simulation (1986), North Holland: North Holland Amsterdam, The Netherlands · Zbl 0603.76101 |
| [48] | Huijben, A. J.M.; Kaasschieter, E. F., Mixed-hybrid finite elements and streamline computation for the potential flow problem, Numer. Methods Partial Differential Equations, 8, 3, 221-266 (1992) · Zbl 0767.76029 |
| [49] | Barry, S. I.; Mercer, G. N., Exact solutions for two-dimensional time-dependent flow and deformation within a poroelastic medium, J. Appl. Mech., 66, 2, 536-540 (1999) |
| [50] | Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018 |
| [51] | Arndt, D.; Bengerth, W.; Clevenger, C.; Davydov, D.; Fehling, M.; Garcia-Sanchez, D.; Harper, G.; Heister, T.; Heltai, L.; Kronbichler, M.; Kynch, R.; Maier, M.; Pelteret, J.-P.; Turcksin, B.; Wells, D., The library, version 9.1, J. Numer. Math. (2019), URL https://dealii.org/deal91-preprint.pdf · Zbl 1435.65010 |
| [52] | Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Dener, A.; Eijkhout, V.; Gropp, W. D.; Karpeyev, D.; Kaushik, D.; Knepley, M. G.; May, D. A.; McInnes, L. C.; Mills, R. T.; Munson, T.; Rupp, K.; Sanan, P.; Smith, B. F.; Zampini, S.; Zhang, H.; Zhang, H., PETSc web page (2019), https://www.mcs.anl.gov/petsc |
| [53] | Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Dener, A.; Eijkhout, V.; Gropp, W. D.; Karpeyev, D.; Kaushik, D.; Knepley, M. G.; May, D. A.; McInnes, L. C.; Mills, R. T.; Munson, T.; Rupp, K.; Sanan, P.; Smith, B. F.; Zampini, S.; Zhang, H.; Zhang, H., PETSc Users ManualTech. Rep. ANL-95/11 - Revision 3.12 (2019), Argonne National Laboratory, URL https://www.mcs.anl.gov/petsc |
| [54] | Ruge, J. W.; Stüben, K., Algebraic multigrid, (Multigrid Methods. Multigrid Methods, Frontiers in Applied Mathematics, vol. 3 (1987), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA), 73-130, Ch. 4 |
| [55] | Falgout, R. D.; Yang, U. M., HYPRE: A library of high performance preconditioners, (Sloot, P. M.A.; Hoekstra, A. G.; Tan, C. J.K.; Dongarra, J. J., Computational Science — ICCS 2002. ICCS 2002. Computational Science — ICCS 2002. ICCS 2002, Lecture Notes in Computer Science, vol. 2331 (2002)), 632-641 · Zbl 1056.65046 |
| [56] | Castelletto, N.; White, J. A.; Tchelepi, H. A., Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics, Int. J. Numer. Anal. Methods Geomech., 39, 1593-1618 (2015) |
| [57] | White, J. A.; Castelletto, N.; Tchelepi, H. A., Block-partitioned solvers for coupled poromechanics: A unified framework, Comput. Methods Appl. Mech. Engrg., 303, 55-74 (2016) · Zbl 1425.74497 |
| [58] | Bergamaschi, L.; Mantica, S.; Manzini, G., A mixed finite element-finite volume formulation of the black oil model, SIAM J. Sci. Comput., 20, 970-997 (1998) · Zbl 0959.76039 |
| [59] | Ferronato, M.; Frigo, M.; Castelletto, N.; White, J. A., Efficient solvers for a stabilized three-field mixed formulation of poroelasticity, (Numerical Mathematics and Advanced Applications. Lecture Notes in Computational Science and Engineering (2020), Springer International Publishing), xxx |
| [60] | Phillips, P. J.; Wheeler, M. F., Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach, Comput. Geosci., 13, 1, 5-12 (2009) · Zbl 1172.74017 |
| [61] | Christie, M. A.; Blunt, M. J., Tenth SPE comparative solution project: A comparison of upscaling techniques, SPE Reserv. Eval. Eng., 4, 4, 308-316 (2001) |
| [62] | White, J. A.; Castelletto, N.; Klevtsov, S.; Bui, Q. M.; Osei-Kuffuor, D.; Tchelepi, H. A., A two-stage preconditioner for multiphase poromechanics in reservoir simulation, Comput. Methods Appl. Mech. Engrg., 357, Article 112575 pp. (2019) · Zbl 1442.76118 |
| [63] | Franceschini, A.; Castelletto, N.; Ferronato, M., Approximate inverse-based block preconditioners in poroelasticity, Comput. Geosci. (2020) |
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.
