An augmented Lagrangian preconditioner for implicitly constituted non-Newtonian incompressible flow. (English) Zbl 1458.65147

The authors are concerned with the construction of a preconditioner for the Newton linearization of a system describing the steady state flow of an implicitly constituted non-Newtonian incompressible fluid. The formulation uses three variables namely, velocity, pressure (mean normal stress) and shear stress and the implicit constitutive relation is of power-law type. The preconditioner is built on a Reynolds-robust preconditioner for the three-dimensional Newtonian system to which the first author contributed. The fundamental idea has been to additionally introduce an augmented Lagrangian term. The preconditioner proved to be robust with respect to variation of the rheological parameters in numerical experiments carried out by authors.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
Full Text: DOI arXiv


[1] F. Abraham, M. Behr, and M. Heinkenschloss, Shape optimization in unsteady blood flow: A numerical study of non-Newtonian effects, Comput. Meth. Biomech. Biomed. Engrg., 8 (2005), pp. 201-212.
[2] M. F. Adams, H. H. Bayraktar, T. M. Keaveny, and P. Papadopoulos, Ultrascalable implicit finite element analyses in solid mechanics with over a half a billion degrees of freedom, in SC ’04: Proceedings of the 2004 ACM/IEEE Conference on Supercomputing, Pittsburgh, PA, USA, 2004, 34, https://doi.org/10.1109/SC.2004.62.
[3] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math., Elsevier, 2003. · Zbl 1098.46001
[4] A. Aposporidis, E. Haber, M. A. Olshanskii, and A. Veneziani, A mixed formulation of the Bingham fluid flow problem: Analysis and numerical solution, Comput. Methods Appl. Mech. Eng., 200 (2011), pp. 2434-2446. · Zbl 1230.76007
[5] C. Bacuta, A unified approach for Uzawa algorithms, SIAM J. Numer. Anal., 44 (2006), pp. 2633-2649, https://doi.org/10.1137/050630714. · Zbl 1128.76052
[6] S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp, D. Karpeyev, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Users Manual, Tech. report ANL-95/11-Revision 3.14, Argonne National Laboratory, 2020, https://www.mcs.anl.gov/petsc.
[7] H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology, Vol. 3, Elsevier, 1989. · Zbl 0729.76001
[8] S. Basu and U. S. Shivhare, Rheological, textural, microstructural, and sensory properties of sorbitol-substituted mango jam, Food Bioprocess Technol., 6 (2013), pp. 1401-1413.
[9] M. Benzi and M. A. Olshanskii, An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput., 28 (2006), pp. 2005-2113, https://doi.org/10.1137/050646421. · Zbl 1126.76028
[10] M. Bercovier and M. Engelman, A finite-element method for incompressible non-Newtonian flows, J. Comput. Phys., 36 (1980), pp. 313-326. · Zbl 0457.76005
[11] R. B. Bird, G. C. Dal, and B. J. Yarusso, The rheology and flow of viscoplastic materials, Rev. Chem. Eng., 1 (1983), pp. 1-70.
[12] J. Blechta, J. Málek, and K. R. Rajagopal, On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion, SIAM J. Math. Anal., 52 (2020), pp. 1232-1289, https://doi.org/10.1137/19M1244895. · Zbl 1432.76075
[13] P. R. Brune, M. G. Knepley, B. F. Smith, and X. Tu, Composing scalable nonlinear algebraic solvers, SIAM Rev., 57 (2015), pp. 535-565, https://doi.org/10.1137/130936725. · Zbl 1336.65030
[14] M. Bulíček, P. Gwiazda, J. Málek, and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., 44 (2012), pp. 2756-2801, https://doi.org/10.1137/110830289. · Zbl 1256.35074
[15] M. Bulíček, P. Gwiazda, J. Málek, and A. Świerczewska-Gwiazda, On steady flows of incompressible fluids with implicit power-law-like rheology, Adv. Calc. Var., 2 (2009), pp. 109-136. · Zbl 1233.35164
[16] E. Burman and A. Linke, Stabilized finite element schemes for incompressible flow using Scott-Vogelius elements, Appl. Numer. Math., 58 (2008), pp. 1704-1719. · Zbl 1148.76031
[17] A. J. Chorin, A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2 (1967), pp. 12-26. · Zbl 0149.44802
[18] L. Diening, C. Kreuzer, and E. Süli, Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, SIAM J. Numer. Anal., 51 (2013), pp. 984-1015, https://doi.org/10.1137/120873133. · Zbl 1268.76030
[19] J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer, 1976, pp. 207-216.
[20] S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996), pp. 16-32, https://doi.org/10.1137/0917003. · Zbl 0845.65021
[21] H. Elman, V. E. Howle, J. Shadid, R. Shuttleworth, and R. Tuminaro, Block preconditioners based on approximate commutators, SIAM J. Sci. Comput., 27 (2006), pp. 1651-1668, https://doi.org/10.1137/040608817. · Zbl 1100.65042
[22] H. Elman and D. Silvester, Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput., 17 (1996), pp. 33-46, https://doi.org/10.1137/0917004. · Zbl 0843.65080
[23] H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford University Press, 2014. · Zbl 1304.76002
[24] R. D. Falgout and U. M. Yang, hypre: A library of high performance preconditioners, in Proceedings of the International Conference on Computational Science (ICCS 2002), Lecture Notes in Comput. Sci. 2331, P. M. A. Sloot, A. G. Hoekstra, C. J. K. Tan, and J. J. Dongarra, eds., Springer, 2002, pp. 632-641. · Zbl 1056.65046
[25] P. E. Farrell, P. A. Gazca-Orozco, and E. Süli, Numerical analysis of unsteady implicitly constituted incompressible fluids: 3-field formulation, SIAM J. Numer. Anal., 58 (2020), pp. 757-787, https://doi.org/10.1137/19M125738X. · Zbl 1434.76065
[26] P. E. Farrell, M. G. Knepley, L. E. Mitchell, and F. Wechsung, PCPATCH: Software for the Topological Construction of Multigrid Relaxation Methods, preprint, https://arxiv.org/abs/1912.08516, 2019, submitted.
[27] P. E. Farrell and J. R. Maddison, Conservative interpolation between volume meshes by local Galerkin projection, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 89-100. · Zbl 1225.76193
[28] P. E. Farrell, L. Mitchell, L. R. Scott, and F. Wechsung, A Reynolds-Robust Preconditioner for the Reynolds-Robust Scott-Vogelius Discretization of the Stationary Incompressible Navier-Stokes Equations, preprint, https://arxiv.org/abs/2004.09398, 2020.
[29] P. E. Farrell, L. Mitchell, L. R. Scott, and F. Wechsung, Robust Multigrid Methods for Nearly Incompressible Elasticity Using Macro Elements, preprint, https://arxiv.org/abs/2002.02051, 2020.
[30] P. E. Farrell, L. Mitchell, and F. Wechsung, An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds number, SIAM J. Sci. Comput., 41 (2019), pp. A3073-A3096, https://doi.org/10.1137/18M1219370. · Zbl 1448.65261
[31] M. Gee, C. Siefert, J. Hu, R. Tuminaro, and M. Sala, ML 5.0 Smoothed Aggregation User’s Guide, Tech. report SAND2006-2649, Sandia National Laboratories, Albuquerque, NM, 2006.
[32] J. W. Glen, The creep of polycrystalline ice, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 228 (1955), pp. 519-538.
[33] R. Glowinski and A. Wachs, On the numerical simulation of viscoplastic fluid flow, in Numerical Methods for Non-Newtonian Fluids, Handb. Numer. Anal. 16, R. Glowinski and J. Xu, eds., Elsevier/North-Holland, 2011, pp. 483-718. · Zbl 1318.76012
[34] P. P. Grinevich and M. A. Olshanskii, An iterative method for the Stokes-type problem with variable viscosity, SIAM J. Sci. Comput., 31 (2009), pp. 3959-3978, https://doi.org/10.1137/08744803. · Zbl 1410.76290
[35] X. He and M. Neytcheva, Preconditioning the incompressible Navier-Stokes equations with variable viscosity, J. Comput. Math., 30 (2012), pp. 461-482. · Zbl 1274.76250
[36] X. He, M. Neytcheva, and C. Vuik, On preconditioning incompressible non-Newtonian flow problems, J. Comput. Math., 33 (2015), pp. 33-58. · Zbl 1349.76011
[37] T. Heister and G. Rapin, Efficient augmented Lagrangian-type preconditioning for the Oseen problem using Grad-Div stabilization, Int. J. Numer. Methods Fluids, 71 (2013), pp. 118-134, https://doi.org/10.1002/fld.3654. · Zbl 1430.76115
[38] V. E. Howle and R. C. Kirby, Block preconditioners for finite element discretization of incompressible flow with thermal convection, Numer. Linear Algebra Appl., 19 (2012), pp. 427-440, https://doi.org/10.1002/nla.1814. · Zbl 1274.76190
[39] J. Hron, J. Málek, J. Stebel, and K. Touška, A Novel View on Computations of Steady Flows of Bingham Fluids Using Implicit Constitutive Relations, preprint MORE/2017/08, Project MORE, Charles University, Prague, 2017.
[40] V. John, A. Linke, C. Merdon, M. Neilan, and L. G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev., 59 (2017), pp. 492-544, https://doi.org/10.1137/15M1047696. · Zbl 1426.76275
[41] D. Kay, D. Loghin, and A. Wathen, A preconditioner for the steady-state Navier-Stokes equations, SIAM J. Sci. Comput., 24 (2002), pp. 237-256, https://doi.org/10.1137/S106482759935808X. · Zbl 1013.65039
[42] Y.-J. Lee, J. Wu, J. Xu, and L. Zikatanov, Robust subspace correction methods for nearly singular systems, Math. Models Methods Appl. Sci., 17 (2007), pp. 1937-1963, https://doi.org/10.1142/S0218202507002522. · Zbl 1151.65096
[43] A. Linke and L. G. Rebholz, Pressure-induced locking in mixed methods for time-dependent (Navier)–Stokes equations, J. Comput. Phys., 388 (2019), pp. 350-356.
[44] K. A. Mardal and R. Winther, Preconditioning discretizations of systems of partial differential equations, Numer. Linear Algebra, 18 (2011), pp. 1-40. · Zbl 1249.65246
[45] S. Matsuhisa and R. B. Bird, Analytical and numerical solutions for laminar flow of the non-Newtonian Ellis fluid, AlChE J., 11 (1965), pp. 588-595.
[46] T. C. Papanastasiou, Flows of materials with yield, J. Rheol., 31 (1987), pp. 385-404. · Zbl 0666.76022
[47] E. C. Pettit and E. D. Waddington, Ice flow at low deviatoric stress, J. Glaciol., 49 (2003), pp. 359-369.
[48] J. Qin, On the Convergence of Some Low Order Mixed Finite Elements for Incompressible Fluids, Ph.D. thesis, Pennsylvania State University, State College, PA, 1994.
[49] K. R. Rajagopal, On implicit constitutive theories, Appl. Math., 48 (2003), pp. 279-319. · Zbl 1099.74009
[50] K. R. Rajagopal, On implicit constitutive theories for fluids, J. Fluid Mech., 550 (2006), pp. 243-249. · Zbl 1097.76009
[51] K. R. Rajagopal and A. R. Srinivasa, On the thermodynamics of fluids defined by implicit constitutive relations, Z. Angew. Math. Phys., 59 (2008), pp. 715-729. · Zbl 1149.76007
[52] F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. T. McRae, G.-T. Bercea, G. R. Markall, and P. H. J. Kelly, Firedrake: Automating the finite element method by composing abstractions, ACM Trans. Math. Software, 43 (2016), 24. · Zbl 1396.65144
[53] D. Sandri, A posteriori estimators for mixed finite element approximations of a fluid obeying the power law, Comput. Methods Appl. Mech. Eng., 166 (1998), pp. 329-340. · Zbl 0953.76057
[54] T. A. Savvas, N. C. Markatos, and C. D. Papaspyrides, On the flow of non-Newtonian polymer solutions, Appl. Math. Model., 18 (1994), pp. 14-22. · Zbl 0800.76039
[55] J. Schöberl, Multigrid methods for a parameter dependent problem in primal variables, Numer. Math., 84 (1999), pp. 97-119. · Zbl 0957.74059
[56] J. Schöberl, Robust Multigrid Methods for Parameter Dependent Problems, Ph.D. thesis, Johannes Kepler Universität Linz, Linz, Austria, 1999. · Zbl 0957.74059
[57] D. Silvester and A. Wathen, Fast iterative solution of stabilised Stokes systems part II: Using general block preconditioners, SIAM J. Numer. Anal., 31 (1994), pp. 1352-1367, https://doi.org/10.1137/0731070. · Zbl 0810.76044
[58] R. Temam, Une méthode d’approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France, 98 (1968), pp. 115-152. · Zbl 0181.18903
[59] T. Tscherpel, FEM for the Unsteady Flow of Implicitly Constituted Incompressible Fluids, Ph.D. thesis, University of Oxford, Oxford, UK, 2018.
[60] J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), pp. 581-613, https://doi.org/10.1137/1034116. · Zbl 0788.65037
[61] J. Xu, The method of subspace corrections, J. Comput. Appl. Math., 128 (2001), pp. 335-362. · Zbl 0983.65133
[62] K. Yasuda, Investigation of the Analogies between Viscometric and Linear Viscoelastic Properties of Polystyrene Fluids, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1979.
[63] S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comput., 74 (2005), pp. 543-554. · Zbl 1085.76042
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.