## Parameter robust preconditioning by congruence for multiple-network poroelasticity.(English)Zbl 1486.65174

### MSC:

 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 65F08 Preconditioners for iterative methods 65F10 Iterative numerical methods for linear systems 76S05 Flows in porous media; filtration; seepage 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74B10 Linear elasticity with initial stresses

### Keywords:

preconditioning; poroelasticity; parameter-robust

poroelasticity
Full Text:

### References:

 [1] M. Bai, D. Elsworth, and J.-C. Roegiers, Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs, Water Res. Res., 29 (1993), pp. 1621-1633. [2] M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), pp. 155-164. · JFM 67.0837.01 [3] J. Brašnová, V. Lukeš, and E. Rohan, Identification of multi-compartment Darcy flow model material parameters, in 20th International Conference on Applied Mechanics 2018, Myslovice, University of West Bohemia, Pilsen, Czech Republic, 2019, pp. 9-13. [4] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Numer. Anal., 8 (1974), pp. 129-151. · Zbl 0338.90047 [5] D. Chou, J. C. Vardakis, L. Guo, B. J. Tully, and Y. Ventikos, A fully dynamic multi-compartmental poroelastic system: Application to aqueductal stenosis, J. Biomech., 49 (2016), pp. 2306-2312. [6] A. Eisenträger and I. Sobey, Multi-fluid poroelastic modelling of CSF flow through the brain, in Poromechanics V: Proceedings of the Fifth Biot Conference on Poromechanics, American Society of Civil Engineers, Reston, VA, 2013, pp. 2148-2157. [7] F. Gaspar, J. Gracia, F. Lisbona, and C. Oosterlee, Distributive smoothers in multigrid for problems with dominating grad-div operators, Numer. Linear Algebra Appl., 15 (2008), pp. 661-683, https://doi.org/10.1002/nla.587. · Zbl 1212.65484 [8] Q. Hong and J. Kraus, Parameter-robust stability of classical three-field formulation of Biot’s consolidation model, Electron. Trans. Numer. Anal., 48 (2018), pp. 202-226, https://doi.org/10.1553/etna_vol48s202. · Zbl 1398.65046 [9] Q. Hong, J. Kraus, M. Lymbery, and F. Philo, Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelasticity models, Numer. Linear Algebra Appl., 26 (2019), e2242. · Zbl 1463.65372 [10] Q. Hong, J. Kraus, M. Lymbery, and F. Philo, 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 (2020), pp. 2523-2555. · Zbl 1471.65143 [11] Q. Hong, J. Kraus, M. Lymbery, and M. F. Wheeler, Parameter-robust convergence analysis of fixed-stress split iterative method for multiple-permeability poroelasticity systems, Multiscale Model. Simul., 18 (2020), pp. 916-941, https://doi.org/10.1137/19M1253988. · Zbl 1447.65077 [12] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 1990. · Zbl 0704.15002 [13] T. Józsa, W. El-Bouri, R. Padmos, S. Payne, and A. Hoekstra, A cerebral circulation model for in silico clinical trials of ischaemic stroke, in The Proceedings of the CompBioMed Conference 2019. [14] J. Lee, A. Cookson, R. Chabiniok, S. Rivolo, E. Hyde, M. Sinclair, C. Michler, T. Sochi, and N. Smith, Multiscale modelling of cardiac perfusion, in Modeling the Heart and the Circulatory System, Springer, Cham, Switzerland, 2015, pp. 51-96. [15] J. J. Lee, Unconditionally Stable Second Order Convergent Partitioned Methods for Multiple-Network Poroelasticity, preprint, https://arxiv.org/abs/1901.06078 (2019). [16] J. J. Lee, K.-A. Mardal, and R. Winther, Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J. Sci. Comput., 39 (2017), pp. A1-A24. · Zbl 1381.76183 [17] J. J. Lee, E. Piersanti, K.-A. Mardal, and M. E. Rognes, A mixed finite element method for nearly incompressible multiple-network poroelasticity, SIAM J. Sci. Comput., 41 (2019), pp. A722-A747. · Zbl 1417.65162 [18] K.-A. Mardal and R. Winther, Preconditioning discretizations of systems of partial differential equations, Numer. Linear Algebra Appl., 18 (2011), pp. 1-40. · Zbl 1249.65246 [19] C. Michler, A. Cookson, R. Chabiniok, E. Hyde, J. Lee, M. Sinclair, T. Sochi, A. Goyal, G. Vigueras, D. Nordsletten, and N. P. Smith, A computationally efficient framework for the simulation of cardiac perfusion using a multi-compartment Darcy porous-media flow model, Int. J. Numer. Methods Biomed. Eng., 29 (2013), pp. 217-232. [20] A. Mikelić and M. F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17 (2013), pp. 455-461, https://doi.org/10.1007/s10596-012-9318-y. · Zbl 1392.35235 [21] M. J. M. Mokhtarudin and S. J. Payne, The study of the function of AQP $$4$$ in cerebral ischaemia-reperfusion injury using poroelastic theory, Int. J. Numer. Methods Biomed. Eng., 33 (2017), e02784. [22] R. Oyarzúa and R. Ruiz-Baier, Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal., 54 (2016), pp. 2951-2973. · Zbl 1457.65210 [23] E. Piersanti, M. E. Rognes, and K.-A. Mardal, Parameter robust preconditioning for multi-compartmental Darcy equations, in EnuMath 2019, Springer, Cham, Switzerland, 2021, pp. 703-711. [24] C. Rodrigo, X. Hu, P. Ohm, J. Adler, F. Gaspar, and L. Zikatanov, New stabilized discretizations for poroelasticity and the Stokes’ equations, Comput. Methods Appl. Mech. Engrg., 341 (2018), pp. 467-484, https://doi.org/10.1016/j.cma.2018.07.003. · Zbl 1440.76027 [25] R. Shipley, P. Sweeney, S. Chapman, and T. Roose, A four-compartment multiscale model of fluid and drug distribution in vascular tumours, Int. J. Numer. Methods Biomed. Eng. 36, (2020), e3315. [26] E. Storvik, J. Both, K. Kuman, J. Nordbotten, and F. Radu, On the optimization of the fixed-stress splitting for Biot’s equations, Internat. J. Numer. Methods. Engrg., 120 (2019), pp. 179-194, https://doi.org/10.1002/nme.6130. [27] K. H. Støverud, M. Aln\aes, H. P. Langtangen, V. Haughton, and K.-A. Mardal, Poro-elastic modeling of syringomyelia-a systematic study of the effects of pia mater, central canal, median fissure, white and gray matter on pressure wave propagation and fluid movement within the cervical spinal cord, Comput. Methods Biomech. Biomed. Eng., 19 (2016), pp. 686-698. [28] B. Tully and Y. Ventikos, Modelling normal pressure hydrocephalus as a “two-hit” disease using multiple-network poroelastic theory, in ASME 2010 Summer Bioengineering Conference, American Society of Mechanical Engineers Digital Collection, Reston, VA. 2013, pp. 877-878. [29] B. J. Tully and Y. Ventikos, Cerebral water transport using multiple-network poroelastic theory: Application to normal pressure hydrocephalus, J. Fluid Mech., 667 (2011), pp. 188-215. · Zbl 1225.76317 [30] J. C. Vardakis, D. Chou, B. J. Tully, C. C. Hung, T. H. Lee, P.-H. Tsui, and Y. Ventikos, Investigating cerebral oedema using poroelasticity, Med. Eng. Phys., 38 (2016), pp. 48-57.
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.