A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations

Farrell, Patrick E.; Mitchell, Lawrence; Scott, L. Ridgway; Wechsung, Florian

doi:10.5802/smai-jcm.72

Farrell, Patrick E.¹ ; Mitchell, Lawrence ² ; Scott, L. Ridgway ³ ; Wechsung, Florian ⁴

¹ Mathematical Institute, University of Oxford, Oxford, UK
² Department of Computer Science, Durham University, Durham, UK
³ Department of Computer Science, University of Chicago, Chicago, USA
⁴ Courant Institute of Mathematical Sciences, New York University, New York, USA

The SMAI Journal of computational mathematics, Tome 7 (2021), pp. 75-96.

Résumé

Augmented Lagrangian preconditioners have successfully yielded Reynolds-robust preconditioners for the stationary incompressible Navier–Stokes equations, but only for specific discretizations. The discretizations for which these preconditioners have been designed possess error estimates which depend on the Reynolds number, with the discretization error deteriorating as the Reynolds number is increased. In this paper we present an augmented Lagrangian preconditioner for the Scott–Vogelius discretization on barycentrically-refined meshes. This achieves both Reynolds-robust performance and Reynolds-robust error estimates. A key consideration is the design of a suitable space decomposition that captures the kernel of the grad-div term added to control the Schur complement; the same barycentric refinement that guarantees inf-sup stability also provides a local decomposition of the kernel of the divergence. The robustness of the scheme is confirmed by numerical experiments in two and three dimensions.

Publié le : 2021-03-24

Zbl

DOI : 10.5802/smai-jcm.72

Classification : 65N55, 65F08, 65N30
Mots clés : Navier–Stokes, Scott–Vogelius element, exactly divergence-free, multigrid, preconditioning, Reynolds-robust solvers

Affiliations des auteurs :

Farrell, Patrick E. ¹ ; Mitchell, Lawrence ² ; Scott, L. Ridgway ³ ; Wechsung, Florian ⁴

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     title = {A {Reynolds-robust} preconditioner for the {Scott-Vogelius} discretization of the stationary incompressible {Navier-Stokes} equations},
     journal = {The SMAI Journal of computational mathematics},
     pages = {75--96},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Farrell, Patrick E.; Mitchell, Lawrence; Scott, L. Ridgway; Wechsung, Florian. A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations. The SMAI Journal of computational mathematics, Tome 7 (2021), pp. 75-96. doi : 10.5802/smai-jcm.72. http://www.numdam.org/articles/10.5802/smai-jcm.72/

Bibliographie
Cité par

[1] Ahmed, N.; Barrenechea, G. R.; Burman, E.; Guzmán, J.; Linke, A.; Merdon, C. A pressure-robust discretization of Oseen’s equation using stabilization in the vorticity equation (2020) (https://arxiv.org/abs/2007.04012)

[2] Arnold, D. N.; Falk, R. S.; Winther, R. Multigrid in $H (div)$ and $H (curl)$ , Numer. Math., Volume 85 (2000) no. 2, pp. 197-217 | DOI | MR | Zbl

[3] Arnold, D. N.; Falk, R.; Winther, R. Preconditioning in $H (div)$ and applications, Math. Comp., Volume 66 (1997) no. 219, pp. 957-984 | DOI | MR | Zbl

[4] Alfeld, P. A trivariate Clough–Tocher scheme for tetrahedral data, Comput. Aided Geom. Des., Volume 1 (1984) no. 2, pp. 169-181 | DOI | Zbl

[5] Ahmed, N.; Linke, A.; Merdon, C. Towards pressure-robust mixed methods for the incompressible Navier–Stokes equations, Comput. Methods Appl. Math., Volume 18 (2018) no. 3, pp. 353-372 | DOI | MR | Zbl

[6] Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; 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 Manual (2019) no. ANL-95/11 - Revision 3.11 (Technical report)

[7] Burman, E.; Fernández, M. A.; Hansbo, P. Continuous interior penalty finite element method for Oseen’s equations, SIAM J. Numer. Anal., Volume 44 (2006) no. 3, pp. 1248-1274 | DOI | MR | Zbl

[8] Burman, E.; Hansbo, P. Edge stabilization for Galerkin approximations of convection–diffusion–reaction problems, Comput. Methods Appl. Mech. Eng., Volume 193 (2004) no. 15, pp. 1437-1453 | DOI | MR | Zbl

[9] Burman, E.; Hansbo, P. Edge stabilization for the generalized Stokes problem: A continuous interior penalty method, Comput. Methods Appl. Mech. Eng., Volume 195 (2006) no. 19, pp. 2393-2410 | DOI | MR | Zbl

[10] Brooks, A. N.; Hughes, T. J. R. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., Volume 32 (1982) no. 1-3, pp. 199-259 | DOI | MR | Zbl

[11] Burman, E.; Linke, A. Stabilized finite element schemes for incompressible flow using Scott–Vogelius elements, Appl. Numer. Math., Volume 58 (2008) no. 11, pp. 1704-1719 | DOI | MR | Zbl

[12] Benzi, M.; Olshanskii, M. A. An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput., Volume 28 (2006) no. 6, pp. 2095-2113 | DOI | MR | Zbl

[13] Benzi, M.; Olshanskii, M. A.; Wang, Z. Modified augmented Lagrangian preconditioners for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, Volume 66 (2011) no. 4, pp. 486-508 | DOI | MR | Zbl

[14] Case, M. A.; Ervin, V. J.; Linke, A.; Rebholz, L. G. A connection between Scott–Vogelius and grad-div stabilized Taylor–Hood FE approximations of the Navier–Stokes equations, SIAM J. Numer. Anal., Volume 49 (2011) no. 4, pp. 1461-1481 | DOI | MR | Zbl

[15] Ciarlet, P. G. The Finite Element Method for Elliptic Problems, North-Holland, 1978 (Reprinted by SIAM in 2002.) | Zbl

[16] Cockburn, B.; Kanschat, G.; Schötzau, D. A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations, J. Sci. Comput., Volume 31 (2007) no. 1-2, pp. 61-73 | DOI | MR | Zbl

[17] Conca, C.; Murat, F.; Pironneau, O. The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Jpn. J. Math., New Ser., Volume 20 (1994) no. 2, pp. 279-318 | DOI | MR | Zbl

[18] Douglas, J.; Dupont, T. Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing Methods in Applied Sciences, Volume 58, Springer, 1976, pp. 207-216 | DOI | MR

[19] Elman, H. C.; Howle, V. E.; Shadid, J.; Shuttleworth, R.; Tuminaro, R. Block preconditioners based on approximate commutators, SIAM J. Sci. Comput., Volume 27 (2006) no. 5, pp. 1651-1668 | DOI | MR | Zbl

[20] Elman, H. C.; Silvester, D. J.; Wathen, A. J. Finite Elements and Fast Iterative Solvers: with applications in incompressible fluid dynamics, Oxford University Press, 2014 | Zbl

[21] Ewing, R. E.; Wang, J. Analysis of the Schwarz algorithm for mixed finite elements methods, ESAIM, Math. Model. Numer. Anal., Volume 26 (1992) no. 6, pp. 739-756 | DOI | Numdam | MR | Zbl

[22] Firedrake-Zenodo Software used in ‘Shape optimisation examples for “Shape Optimisation and Robust Solvers for Incompressible Flow”’, 2019 (https://zenodo.org/record/3369183) | DOI

[23] Fu, G.; Guzmán, J.; Neilan, M. Exact smooth piecewise polynomial sequences on Alfeld splits, Math. Comp., Volume 89 (2020) no. 323, pp. 1059-1091 | DOI | MR | Zbl

[24] Farrell, P. E.; Knepley, M. G.; Mitchell, L.; Wechsung, F. PCPATCH: Software for the topological construction of multigrid relaxation methods (2021) (https://arxiv.org/abs/1912.08516, to appear in ACM Trans. Math. Softw.)

[25] Farrell, P. E.; Mitchell, L.; Scott, L. R.; Wechsung, F. florianwechsung/alfi: Reynolds-Robust Preconditioners for the NSE, 2020 (https://doi.org/10.5281/zenodo.4311759) | DOI

[26] Farrell, P. E.; Mitchell, L.; Scott, L. R.; Wechsung, F. Robust multigrid methods for nearly incompressible elasticity using macro elements, 2020 (https://arxiv.org/abs/2002.02051)

[27] Farrell, P. E.; Mitchell, L.; Wechsung, F. An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier–Stokes equations at high Reynolds number, SIAM J. Sci. Comput., Volume 41 (2019) no. 5, p. A3073-A3096 | arXiv | DOI | MR | Zbl

[28] Gelhard, T.; Lube, G.; Olshanskii, M. A.; Starcke, J.-H. Stabilized finite element schemes with LBB-stable elements for incompressible flows, J. Comput. Appl. Math., Volume 177 (2005) no. 2, pp. 243-267 | DOI | MR | Zbl

[29] Gauger, N. R.; Linke, A.; Schroeder, P. W. On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond, The SMAI Journal of Computational Mathematics, Volume 5 (2019), pp. 89-129 | DOI | Numdam | MR | Zbl

[30] Geuzaine, C.; Remacle, J.-F. Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Meth. Engng., Volume 79 (2009) no. 11, pp. 1309-1331 | DOI | MR | Zbl

[31] Hong, Q.; Kraus, J.; Xu, J.; Zikatanov, L. A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations, Numer. Math., Volume 132 (2015) no. 1, pp. 23-49 | DOI | MR

[32] Heister, T.; Rapin, G. Efficient augmented Lagrangian-type preconditioning for the Oseen problem using Grad-Div stabilization, International Journal for Numerical Methods in Fluids, Volume 71 (2012) no. 1, pp. 118-134 | DOI | MR | Zbl

[33] John, V.; Linke, A.; Merdon, C.; Neilan, M.; Rebholz, L. G. On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev., Volume 59 (2017) no. 3, pp. 492-544 | DOI | MR | Zbl

[34] Kay, D.; Loghin, D.; Wathen, A. J. A preconditioner for the steady-state Navier–Stokes equations, SIAM J. Sci. Comput., Volume 24 (2002) no. 1, pp. 237-256 | DOI | MR

[35] Kolesnikov, A.; Sorokina, T. Multivariate $C^{1}$ -Continuous Splines on the Alfeld Split of a Simplex, Approximation Theory XIV: San Antonio 2013, Volume 83, Springer, 2014, pp. 283-294 (Series Title: Springer Proceedings in Mathematics & Statistics) | DOI | MR | Zbl

[36] Li, X. S.; Demmel, J. W. SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Trans. Math. Softw., Volume 29 (2003) no. 2, pp. 110-140 | DOI | Zbl

[37] Li, X. S.; Demmel, J. W.; Gilbert, J. R.; Grigori, L.; Shao, M.; Yamazaki, I. SuperLU Users’ Guide (1999) no. LBNL-44289 (Technical report)

[38] Landau, L. D.; Lifshitz, E. M. Fluid mechanics, Course of Theoretical Physics, Elsevier/Butterworth Heinemann, 1987 no. 6 | Zbl

[39] Linke, A.; Rebholz, L. G. Pressure-induced locking in mixed methods for time-dependent (Navier–)Stokes equations, J. Comput. Phys., Volume 388 (2019), pp. 350-356 | DOI | MR | Zbl

[40] Lehrenfeld, C.; Schöberl, J. High order exactly divergence-free hybrid discontinuous Galerkin Methods for unsteady incompressible flows, Comput. Methods Appl. Mech. Eng., Volume 307 (2016), pp. 339-361 | DOI | MR | Zbl

[41] Lee, Y.-J.; Wu, J.; Chen, J. Robust multigrid method for the planar linear elasticity problems, Numer. Math., Volume 113 (2009) no. 3, pp. 473-496 | DOI | MR

[42] Lee, Y.-J.; Wu, J.; Xu, J.; Zikatanov, L. Robust subspace correction methods for nearly singular systems, Math. Models Methods Appl. Sci., Volume 17 (2007) no. 11, pp. 1937-1963 | DOI | MR | Zbl

[43] May, D. A.; Sanan, P.; Rupp, K.; Knepley, M. G.; Smith, B. F. Extreme-scale multigrid components within PETSc, Proceedings of the Platform for Advanced Scientific Computing Conference, ACM (2016) | DOI

[44] Neilan, M. The Stokes complex: a review of exactly divergence–free finite element pairs for incompressible flows, 75 Years of Mathematics of Computation (Brenner, S. C.; Shparlinksi, I.; Shu, C. W.; Szyld, D. B., eds.) (Contemporary Mathematics), Volume 754, American Mathematical Society, 2020, pp. 141-158 | DOI | MR

[45] Olshanskii, M. A.; Rebholz, L. G. Application of barycenter refined meshes in linear elasticity and incompressible fluid dynamics, Electron. Trans. Numer. Anal., Volume 38 (2011), pp. 258-274 | MR | Zbl

[46] Patankar, S. Numerical Heat Transfer and Fluid Flow, Hemisphere Series on Computational Methods in Mechanics and Thermal Science, Taylor & Francis, 1980 | Zbl

[47] Powell, M. J. D.; Sabin, M. A. Piecewise quadratic approximations on triangles, ACM Trans. Math. Softw., Volume 3 (1977) no. 4, pp. 316-325 | DOI | MR | Zbl

[48] Qin, J. On the convergence of some low order mixed finite elements for incompressible fluids, Ph. D. Thesis, Pennsylvania State University (1994)

[49] Quarteroni, A.; Valli, A. Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23, Springer, 2008 | Zbl

[50] Rathgeber, F.; Ham, D. A.; Mitchell, L.; Lange, M.; Luporini, F.; Mcrae, A. T. T.; Bercea, G.-T.; Markall, G. R.; Kelly, P. H. J. Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Softw., Volume 43 (2016) no. 3, pp. 1-27 | DOI | MR | Zbl

[51] Saad, Y. A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., Volume 14 (1993) no. 2, pp. 461-469 | DOI | MR | Zbl

[52] Schöberl, J. Robust Multigrid Preconditioning for Parameter-Dependent Problems I: The Stokes-Type Case, Multigrid Methods V (Lecture Notes in Computational Science and Engineering), Springer (1998), pp. 260-275 | DOI | MR | Zbl

[53] Schöberl, J. Multigrid methods for a parameter dependent problem in primal variables, Numer. Math., Volume 84 (1999) no. 1, pp. 97-119 | DOI | MR | Zbl

[54] Schöberl, J. Robust Multigrid Methods for Parameter Dependent Problems, Ph. D. Thesis, Johannes Kepler Universität Linz (1999)

[55] Shih, T. M.; Tan, C. H.; Hwang, B. C. Effects of grid staggering on numerical schemes, International Journal for Numerical Methods in Fluids, Volume 9 (1989) no. 2, pp. 193-212 | DOI | Zbl

[56] Scott, L. R.; Vogelius, M. Conforming finite element methods for incompressible and nearly incompressible continua, Large Scale Computations in Fluid Mechanics, B. E. Engquist, ıt et al., eds., Volume 22 (Part 2), Providence: AMS (1985), pp. 221-244 | Zbl

[57] Scott, L. R.; Vogelius, M. Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, ESAIM, Math. Model. Numer. Anal., Volume 19 (1985) no. 1, pp. 111-143 | DOI | Numdam | MR | Zbl

[58] Turek, S. Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach, Lecture Notes in Computational Science and Engineering, Springer, 1999 | Zbl

[59] Vanka, S. P. Block-implicit multigrid solution of Navier-Stokes equations in primitive variables, J. Comput. Phys., Volume 65 (1986) no. 1, pp. 138-158 | DOI | MR | Zbl

[60] Wechsung, F. florianwechsung/alfi: Thesis submission, 2019 (https://doi.org/10.5281/zenodo.3369207) | DOI

[61] Worsey, A. J.; Farin, G. An $n$ -dimensional Clough–Tocher interpolant, Computing, Volume 3 (1987) no. 1, pp. 99-110 | DOI | MR | Zbl

[62] Wang, J.; Ye, X. New finite element methods in computational fluid dynamics by $H (div)$ elements, SIAM J. Numer. Anal., Volume 45 (2007) no. 3, pp. 1269-1286 | DOI | MR

[63] Xu, J. Iterative methods by space decomposition and subspace correction, SIAM Rev., Volume 34 (1992) no. 4, pp. 581-613 | DOI | MR | Zbl

[64] Software used in ‘A Reynolds-robust preconditioner for the Reynolds-robust Scott–Vogelius discretization of the stationary incompressible Navier–Stokes equations’, 2020 (https://zenodo.org/record/4311750) | DOI

[65] Zhang, S. A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp., Volume 74 (2004) no. 250, pp. 543-555 | DOI | MR

[66] Zhang, S. On the $P 1$ Powell–Sabin divergence free finite element for the Stokes equations, Journal of Computational Mathematics, Volume 26 (2008) no. 3, pp. 456-470 | MR | Zbl

[67] Zhang, S. Divergence-free finite elements on tetrahedral grids for k $\geq$ 6, Math. Comp., Volume 80 (2011) no. 274, pp. 669-695 | DOI | MR | Zbl

[68] Zhang, S. Quadratic divergence-free finite elements on Powell–Sabin tetrahedral grids, Calcolo, Volume 48 (2011) no. 3, pp. 211-244 | DOI | MR | Zbl

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