A block-diagonal preconditioner for a four-field mixed finite element method for Biot’s equations. (English) Zbl 1433.76069
Summary: In this paper, we explore an efficient preconditioning method for the saddle point system resulting from a four-field mixed finite element method applied to Biot’s consolidation model. The proposed preconditioner is a block diagonal preconditioner based on the Schur complement. We obtain bounds on the eigenvalues of the preconditioned matrix that are clustered away from 0. To reduce the computational expense, this preconditioner is inverted approximately. Some numerical results are provided to show the efficiency of our preconditioning strategy when applied to a poroelasticity problem in a layered medium.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 65F08 | Preconditioners for iterative methods |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Software:
OctaveReferences:
| [1] | Arnold, D. N.; Falk, R. S.; Winther, R., Preconditioning in \(H(div)\) and applications, Math. Comput., 66, 219, 957-984 (1997) · Zbl 0870.65112 |
| [2] | Benzi, M.; Wathen, A. J., Some preconditioning techniques for saddle point problems, (Model Order Reduction: Theory, Research Aspects and Applications. Model Order Reduction: Theory, Research Aspects and Applications, Math. Ind., vol. 13 (2008), Springer: Springer Berlin), 195-211 · Zbl 1152.65425 |
| [3] | Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 2, 155-164 (1941) · JFM 67.0837.01 |
| [4] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer · Zbl 0788.73002 |
| [5] | (2012), (Online; accessed 4 November 2014) |
| [6] | Chen, S.-C.; Wang, Y.-N., Conforming rectangular mixed finite elements for elasticity, J. Sci. Comput., 47, 1, 93-108 (2011) · Zbl 1248.74038 |
| [7] | Cheung, Y. K.; Tham, L. G., Numerical solutions for Biot’s consolidation of layered soil, J. Eng. Mech., 109, 3, 669-679 (1983) |
| [8] | Cowin, S. C., Bone poroelasticity, J. Biomech., 32, 3, 217-238 (1999) |
| [9] | Eaton, J. W.; Bateman, D.; Hauberg, S., GNU Octave version 3.0.1 manual: a high-level interactive language for numerical computations (2009), (Online; accessed 1 Feburary 2017) |
| [10] | Elman, H. C.; Silvester, D. J.; Wathen, A. J., Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics (2005), Oxford University Press · Zbl 1083.76001 |
| [11] | George, A.; Ng, E., On the complexity of sparse QR and LU factorization of finite-element matrices, SIAM J. Sci. Stat. Comput., 9, 5, 849-861 (1988) · Zbl 0658.65023 |
| [12] | Gurevich, B.; Schoenberg, M., Interface conditions for Biot’s equations of poroelasticity, J. Acoust. Soc. Am., 105, 5, 2585-2589 (1999) |
| [13] | 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) |
| [14] | Hellinger, E., Die allgemeinen ansätze der mechanik der kontinua, Grundlehren Math. Wiss., 30, 602-694 (1914) · JFM 45.1012.01 |
| [15] | Hudson, J. A.; Stephansson, O.; Andersson, J.; Tsang, C.-F.; Jing, L., Coupled T-H-M issues relating to radioactive waste repository design and performance, Int. J. Rock Mech. Min. Sci., 38, 1, 143-161 (2001) |
| [16] | Lewis, R. W.; Schrefler, B. A., The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media (1998), John Wiley · Zbl 0935.74004 |
| [17] | Lipnikov, K., Numerical Methods for the Biot Model in Poroelasticity (2002), University of Houston, PhD thesis |
| [18] | Marchuk, G. I.; Kuznetsov, Y. A., On optimal iteration processes, Sov. Math. Dokl., 9, 4, 1041-1045 (1968) · Zbl 0179.21406 |
| [19] | Morris, J. P.; McNab, W. W.; Carroll, S. K.; Hao, Y.; Foxall, W.; Wagoner, J. L., Injection and Reservoir Hazard Management: the Role of Injection-Induced Mechanical Deformation and Geochemical Alteration at in Salah CO2 Storage Project (2009), status report quarter end, June 2009. Technical report, LLNL Technical Report |
| [20] | Murphy, M. F.; Golub, G. H.; Wathen, A. J., A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21, 6, 1969-1972 (2000) · Zbl 0959.65063 |
| [21] | 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, 2, 131-144 (2007) · Zbl 1117.74015 |
| [22] | 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, 2, 145-158 (2007) · Zbl 1117.74016 |
| [23] | Phillips, P. J.; Wheeler, M. F., A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci., 12, 4, 417-435 (2008) · Zbl 1155.74048 |
| [24] | 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 |
| [25] | Rajapakse, R., Stress analysis of borehole in poroelastic medium, J. Eng. Mech., 119, 6, 1205-1227 (1993) |
| [26] | Raju, M.; Khaitan, S. K., High performance computing of three-dimensional finite element codes on a 64-bit machine, J. Appl. Fluid Mech., 5, 2, 123-132 (2012) |
| [27] | Raviart, P.-A.; Thomas, J.-M., A mixed finite element method for 2-nd order elliptic problems, (Mathematical Aspects of Finite Element Methods (1977), Springer), 292-315 · Zbl 0362.65089 |
| [28] | Roose, T.; Netti, P. A.; Munn, L. L.; Boucher, Y.; Jain, R. K., Solid stress generated by spheroid growth estimated using a linear poroelasticity model, Microvasc. Res., 66, 3, 204-212 (2003) |
| [29] | Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM · Zbl 1002.65042 |
| [30] | Settari, A.; Walters, D. A.; Behie, G. A., Use of coupled reservoir and geomechanical modelling for integrated reservoir analysis and management, J. Can. Pet. Technol., 40, 12 (2001) |
| [31] | Showalter, R. E., Diffusion in poro-elastic media, J. Math. Anal. Appl., 251, 1, 310-340 (2000) · Zbl 0979.74018 |
| [32] | Tang, X.-W.; Onitsuka, K., Consolidation of double-layered ground with vertical drains, Int. J. Numer. Anal. Methods Geomech., 25, 14, 1449-1465 (2001) · Zbl 1112.74437 |
| [33] | Toh, K.-C.; Phoon, K.-K.; Chan, S.-H., Block preconditioners for symmetric indefinite linear systems, Int. J. Numer. Methods Eng., 60, 8, 1361-1381 (2004) · Zbl 1065.65064 |
| [34] | Trefethen, L. N.; Bau, D., Numerical Linear Algebra, vol. 50 (1997), SIAM · Zbl 0874.65013 |
| [35] | Wan, J., Stabilized Finite Element Methods for Coupled Geomechanics and Multiphase Flow (2002), Stanford University, PhD thesis |
| [36] | Wang, H. F., Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology (2000), Princeton University Press |
| [37] | Yi, S.-Y., Convergence analysis of a new mixed finite element method for Biot’s consolidation model, Numer. Methods Partial Differ. Equ., 30, 4, 1189-1210 (2014) · Zbl 1350.74024 |
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.
