×

Parameter-robust Uzawa-type iterative methods for double saddle point problems arising in Biot’s consolidation and multiple-network poroelasticity models. (English) Zbl 1471.65143

The authors consider the stationary iterative methods for solving the equations of multiple-network poroelastic theory which describe flow in deformable porous media. They propose and analyze a class of fully decoupled iterative schemes. The proposed method fully decouples the fluid velocity, fluid pressure and solid displacement fields, contrary to the fixed-stress iterative scheme, which decouples only the flow from the mechanics problem. In every iteration, the smaller subsystems are solved. The authors present convergence analysis which proves the parameter-robust linear convergence of the new algorithm. The rate of contraction is strictly less than one independent of all physical and discretization parameters. The theoretical results are confirmed by a series of numerical tests.
Reviewer: Yan Xu (Hefei)

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
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adler, J., Gaspar, F., Hu, X., Rodrigo, C. and Zikatanov, L., Robust block preconditioners for Biot’s model, in Domain Decomposition Methods in Science and Engineering XXIV. DD 2017, , Vol. 125 (Springer, 2019), pp. 3-16. · Zbl 1442.65342
[2] Almani, T., Kumar, K., Dogru, A., Singh, G. and Wheeler, M., Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics, Comput. Methods Appl. Mech. Engrg.311 (2016) 180-207. · Zbl 1439.74183
[3] Almani, T., Kumar, K. and Wheeler, M., Convergence and error analysis of fully discrete iterative coupling schemes for coupling flow with geomechanics, Comput. Geosci.21 (2017) 1157-1172. · Zbl 1396.76047
[4] Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M. and Wells, G., The FEniCS project version 1.5, Arch. Numer. Softw.3 (2015) 9-23.
[5] Arnold, D., Falk, R. and Winther, R., Preconditioning in H(div) and applications, Math. Comp.66 (1997) 957-984. · Zbl 0870.65112
[6] Bærland, T., Lee, J., Mardal, K.-A. and Winther, R., Weakly imposed symmetry and robust preconditioners for Biot’s consolidation model, Comput. Methods Appl. Math.17 (2017) 377-396. · Zbl 1421.74095
[7] Bai, M., Elsworth, D. and Roegiers, J.-C., Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs, Water Resour. Res.29 (1993) 1621-1633.
[8] Barenblatt, G., Zheltov, G. and Kochina, I., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], J. Appl. Math. Mech.24 (1960) 1286-1303. · Zbl 0104.21702
[9] Bause, M., Radu, F. and Köcher, U., Space-time finite element approximation of the Biot poroelasticity system with iterative coupling, Comput. Methods Appl. Mech. Engrg.320 (2017) 745-768. · Zbl 1439.74389
[10] Benzi, M. and Beik, F., Iterative methods for double saddle point systems, SIAM J. Matrix Anal. Appl.39 (2018) 902-921. · Zbl 1391.65062
[11] Benzi, M. and Beik, F., Uzawa-type and augmented Lagrangian methods for double saddle point systems, in Structured Matrices in Numerical Linear Algebra, eds. Bini, P. D. A., Benedetto, P. F. D., Tyrtyshnikov, P. E. and Barel, P. M. V. (Springer International Publishing, 2019), Chap. 11.
[12] Biot, M., General theory of three-dimensional consolidation, J. Appl. Phys.12 (1941) 155-164. · JFM 67.0837.01
[13] Biot, M., Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys.26 (1955) 182-185. · Zbl 0067.23603
[14] Boffi, D., Botti, M. and Di Pietro, D., A nonconforming high-order method for the Biot problem on general meshes, SIAM J. Sci. Comput.38 (2016) A1508-A1537. · Zbl 1337.76042
[15] Boffi, D., Brezzi, F. and Fortin, M., Mixed Finite Element Methods and Applications, , Vol. 44 (Springer, 2013). · Zbl 1277.65092
[16] Both, J. W., Borregales, M., Nordbotten, J., Kumar, K. and Radu, F., Robust fixed stress splitting for Biot’s equations in heterogeneous media, Appl. Math. Lett.68 (2017) 101-108. · Zbl 1383.74025
[17] Both, J., Kumar, K., Nordbotten, J. and Radu, F., Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media, Comput. Math. Appl.77 (2019) 1479-1502. · Zbl 1442.65251
[18] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Fr. Automat. Inform. Rech. Opér. Sér. Rouge8 (1974) 129-151. · Zbl 0338.90047
[19] Brun, M. K., Ahmed, E., Berre, I., Nordbotten, J. M. and Radu, F. A., Monolithic and splitting based solution schemes for fully coupled quasi-static thermo-poroelasticity with nonlinear convective transport, Comput. Math. Appl.80 (2020) 1964-1984. · Zbl 1451.74204
[20] Chou, D., Vardakis, J., Guo, L., Tully, B. and Ventikos, Y., A fully dynamic multi-compartmental poroelastic system: Application to aqueductal stenosis, J. Biomech.49 (2016) 2306-2312.
[21] Costabel, M. and Dauge, M., On the inequalities of Babuška-Aziz, Friedrichs and Horgan-Payne, Arch. Ration. Mech. Anal.217 (2015) 873-898. · Zbl 1329.35019
[22] Dana, S. and Wheeler, M., Convergence analysis of two-grid fixed stress iterative scheme for coupled flow and deformation in heterogeneous poroelastic media, Comput. Methods Appl. Mech. Engrg.341 (2018) 788-806. · Zbl 1440.74122
[23] L. Guo, Z. Li, J. Lyu, Y. Mei, J. Vardakis, D. Chen, C. Han, X. Lou and Y. Ventikos, On the validation of a multiple-network poroelastic model using arterial spin labeling MRI data, Front. Comput. Neurosci.13.
[24] 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. and Ventikos, Y., Subject-specific multi-poroelastic model for exploring the risk factors associated with the early stages of Alzheimer’s disease, Interface Focus8 (2018) 20170019.
[25] Hiptmair, R. and Xu, J., Nodal auxiliary space preconditioning in \(H(\text{curl})\) and \(H(\text{div})\) spaces, SIAM J. Numer. Anal.45 (2007) 2483-2509(electronic). · Zbl 1153.78006
[26] Hong, Q. and Kraus, J., Parameter-robust stability of classical three-field formulation of Biot’s consolidation model, Electron. Trans. Numer. Anal.48 (2018) 202-226. · Zbl 1398.65046
[27] Hong, Q., Kraus, J., Lymbery, M. and Philo, F., Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelasticity models, Numer. Linear Algebra Appl.26 (2019) e2242, arXiv:1806.00353v2. · Zbl 1463.65372
[28] Hong, Q., Kraus, J., Lymbery, M. and Wheeler, M. F., Parameter-robust convergence analysis of fixed-stress split iterative method for multiple-permeability poroelasticity systems, Multiscale Model. Simul.18 (2020) 916-941. · Zbl 1447.65077
[29] Hong, Q., Kraus, J., Xu, J. and Zikatanov, L., A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations, Numer. Math.132 (2016) 23-49. · Zbl 1338.76054
[30] Hu, X., Rodrigo, C., Gaspar, F. and Zikatanov, L., A nonconforming finite element method for the Biot’s consolidation model in poroelasticity, J. Comput. Appl. Math.310 (2017) 143-154. · Zbl 1381.76175
[31] Kanschat, G. and Riviere, B., A finite element method with strong mass conservation for Biot’s linear consolidation model, J. Sci. Comput.77 (2018) 1762-1779. · Zbl 1407.65192
[32] J. Kim, H. Tchelepi and R. Juanes, Stability, accuracy and efficiency of sequential methods for coupled flow and geomechanics, SPE J.16.
[33] Kolesov, A. and Vabishchevich, P., Splitting schemes with respect to physical processes for double-porosity poroelasticity problems, Russ. J. Numer. Anal. Math. Model.32 (2017) 99-113. · Zbl 1457.76100
[34] Kraus, J., Lazarov, R., Lymbery, M., Margenov, S. and Zikatanov, L., Preconditioning heterogeneous H(div) problems by additive Schur complement approximation and applications, SIAM J. Sci. Comput.38 (2016) A875-A898. · Zbl 1380.65376
[35] Lee, J., Robust error analysis of coupled mixed methods for Biot’s consolidation model, J. Sci. Comput.69 (2016) 610-632. · Zbl 1368.65234
[36] Lee, J., Mardal, K.-A. and Winther, R., Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J. Sci. Comput.39 (2017) A1-A24. · Zbl 1381.76183
[37] Lee, J., Piersanti, E., Mardal, K.-A. and Rognes, M., A mixed finite element method for nearly incompressible multiple-network poroelasticity, SIAM J. Sci. Comput.41 (2019) A722-A747. · Zbl 1417.65162
[38] Lee, Y., Wu, J. and Chen, J., Robust multigrid method for the planar linear elasticity problems, Numer. Math.113 (2009) 473-496. · Zbl 1170.74050
[39] Lee, Y. J., Wu, J., Xu, L. and Zikatanov, J., Robust subspace correction methods for nearly singular systems, Math. Models Methods Appl. Sci.17 (2007) 1937-1963. · Zbl 1151.65096
[40] Lee, Y., Wu, J., Xu, J. and Zikatanov, L., A sharp convergence estimate for the method of subspace corrections for singular systems of equations, Math. Comp.77 (2008) 831. · Zbl 1137.65058
[41] Logg, A., Mardal, K.-A., Wells, G.et al., Automated Solution of Differential Equations by the Finite Element Method (Springer, 2012). · Zbl 1247.65105
[42] Mardal, K.-A. and Winther, R., Preconditioning discretizations of systems of partial differential equations, Numer. Linear Algebra Appl.18 (2011) 1-40. · Zbl 1249.65246
[43] A. Mikelić and M. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci.17. · Zbl 1392.35235
[44] N. A. for Finite element methods & standards (Great Britain), The Standard NAFEMS Benchmarks (NAFEMS, 1990).
[45] Oyarzúa, R. and Ruiz-Baier, R., Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal.54 (2016) 2951-2973. · Zbl 1457.65210
[46] Rodrigo, C., Hu, X., Ohm, P., Adler, J., Gaspar, F. and Zikatanov, L., New stabilized discretizations for poroelasticity and the Stokes’ equations, Comput. Methods Appl. Mech. Eng.341 (2018) 467-484. · Zbl 1440.76027
[47] Schöberl, J., Multigrid methods for a parameter dependent problem in primal variables, Numer. Math.84 (1999) 97-119. · Zbl 0957.74059
[48] Showalter, R., Poroelastic filtration coupled to Stokes flow, Lecture Notes in Pure and Appl. Math.242 (2010) 229-241. · Zbl 1084.76070
[49] Sogn, J. and Zulehner, W., Schur complement preconditioners for multiple saddle point problems of block tridiagonal form with application to optimization problems, IMA J. Numer. Anal.39 (2019) 1328-1359. · Zbl 1464.65032
[50] Storvik, E., Both, J., Kumar, K., Nordbotten, J. and Radu, F., On the optimization of the fixed-stress splitting for Biot’s equations, Int. J. Numer. Methods Engrg.120 (2019) 179-194.
[51] Tully, B. and Ventikos, Y., Cerebral water transport using multiple-network poroelastic theory: Application to normal pressure hydrocephalus, J. Fluid Mech.667 (2011) 188-215. · Zbl 1225.76317
[52] Vardakis, J., Chou, D., Tully, B., Hung, C., Lee, T., Tsui, P. and Ventikos, Y., Investigating cerebral oedema using poroelasticity, Med. Eng. Phys.38 (2016) 48-57.
[53] J. C. Vardakis, L. Guo, T. W. Peach, T. Lassila, M. Mitolo, D. Chou et al., Fluid-structure interaction for highly complex, statistically defined, biological media: Homogenisation and a 3d multi-compartmental poroelastic model for brain biomechanics, J. Fluids Struct.91.
[54] Vardakis, J., Tully, B. and Ventikos, Y., Exploring the efficacy of endoscopic ventriculostomy for hydrocephalus treatment via a multicompartmental poroelastic model of CSF transport: A computational perspective, PLoS ONE8 (2013) e84577.
[55] Vassilevski, P. S. and Lazarov, R. D., Preconditioning mixed finite element saddle-point elliptic problems, Numer. Linear Algebra Appl.3 (1996) 1-20. · Zbl 0848.65079
[56] White, J., Castelletto, N. and Tchelepi, H., Block-partitioned solvers for coupled poromechanics: A unified framework, Comput. Methods Appl. Mech. Engrg.303 (2016) 55-74. · Zbl 1425.74497
[57] Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Rev.34 (1992) 581-613. · Zbl 0788.65037
[58] Xu, J. and Zikatanov, L., The method of alternating projections and the method of subspace corrections in Hilbert space, J. Amer. Math. Soc.15 (2002) 573-597. · Zbl 0999.47015
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.