On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond
Gauger, Nicolas R.1 ; Linke, Alexander2 ; Schroeder, Philipp W.3
1 Chair for Scientific Computing, TU Kaiserslautern, 67663 Kaiserslautern, Germany
2 Weierstrass Institute, 10117 Berlin, Germany
3 Institute for Numerical and Applied Mathematics, Georg-August-Universität Göttingen, 37083 Göttingen, Germany
The SMAI Journal of computational mathematics, Tome 5 (2019), pp. 89-129.

An improved understanding of the divergence-free constraint for the incompressible Navier–Stokes equations leads to the observation that a semi-norm and corresponding equivalence classes of forces are fundamental for their nonlinear dynamics. The recent concept of pressure-robustness allows to distinguish between space discretisations that discretise these equivalence classes appropriately or not. This contribution compares the accuracy of pressure-robust and non-pressure-robust space discretisations for transient high Reynolds number flows, starting from the observation that in generalised Beltrami flows the nonlinear convection term is balanced by a strong pressure gradient. Then, pressure-robust methods are shown to outperform comparable non-pressure-robust space discretisations. Indeed, pressure-robust methods of formal order k are comparably accurate than non-pressure-robust methods of formal order 2k on coarse meshes. Investigating the material derivative of incompressible Euler flows, it is conjectured that strong pressure gradients are typical for non-trivial high Reynolds number flows. Connections to vortex-dominated flows are established. Thus, pressure-robustness appears to be a prerequisite for accurate incompressible flow solvers at high Reynolds numbers. The arguments are supported by numerical analysis and numerical experiments.

Publié le :
DOI : 10.5802/smai-jcm.44
Classification : 65M12, 65M15, 65M60, 76D05, 76D10, 76D17
Mots clés : incompressible Navier–Stokes, pressure-robust methods, Helmholtz–Hodge projector, Discontinuous Galerkin method, divergence-free $H$(div) finite elements, structure-preserving algorithms, high-order methods, (generalised) Beltrami flows, high Reynolds number flows, material derivative
Gauger, Nicolas R. 1 ; Linke, Alexander 2 ; Schroeder, Philipp W. 3

1 Chair for Scientific Computing, TU Kaiserslautern, 67663 Kaiserslautern, Germany
2 Weierstrass Institute, 10117 Berlin, Germany
3 Institute for Numerical and Applied Mathematics, Georg-August-Universität Göttingen, 37083 Göttingen, Germany 
@article{SMAI-JCM_2019__5__89_0,
     author = {Gauger, Nicolas R. and Linke, Alexander and Schroeder, Philipp W.},
     title = {On high-order pressure-robust space discretisations, their advantages for incompressible high {Reynolds} number generalised {Beltrami} flows and beyond},
     journal = {The SMAI Journal of computational mathematics},
     pages = {89--129},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {5},
     year = {2019},
     doi = {10.5802/smai-jcm.44},
     zbl = {07090176},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/smai-jcm.44/}
}
TY  - JOUR
AU  - Gauger, Nicolas R.
AU  - Linke, Alexander
AU  - Schroeder, Philipp W.
TI  - On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond
JO  - The SMAI Journal of computational mathematics
PY  - 2019
SP  - 89
EP  - 129
VL  - 5
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - http://www.numdam.org/articles/10.5802/smai-jcm.44/
DO  - 10.5802/smai-jcm.44
LA  - en
ID  - SMAI-JCM_2019__5__89_0
ER  - 
%0 Journal Article
%A Gauger, Nicolas R.
%A Linke, Alexander
%A Schroeder, Philipp W.
%T On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond
%J The SMAI Journal of computational mathematics
%D 2019
%P 89-129
%V 5
%I Société de Mathématiques Appliquées et Industrielles
%U http://www.numdam.org/articles/10.5802/smai-jcm.44/
%R 10.5802/smai-jcm.44
%G en
%F SMAI-JCM_2019__5__89_0
Gauger, Nicolas R.; Linke, Alexander; Schroeder, Philipp 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, Tome 5 (2019), pp. 89-129. doi : 10.5802/smai-jcm.44. http://www.numdam.org/articles/10.5802/smai-jcm.44/

[1] Ahmed, N.; Linke, A.; Merdon, C. On really locking-free mixed finite element methods for the transient incompressible Stokes equations, SIAM J. Numer. Anal., Volume 56 (2018) no. 1, pp. 185-209 | MR | Zbl

[2] 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 | MR | Zbl

[3] Akbas, M.; Linke, A.; Rebholz, L. G.; Schroeder, P. W. The analogue of grad-div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes, Comput. Methods Appl. Mech. Engrg., Volume 341 (2018), pp. 917-938 | DOI | MR

[4] Arndt, D.; Dallmann, H.; Lube, G. Local projection FEM stabilization for the time-dependent incompressible Navier–Stokes problem, Numer. Methods Partial Differential Equations, Volume 31 (2015) no. 4, pp. 1224-1250 | MR | Zbl

[5] Arnold, D. N.; Brezzi, F.; Fortin, M. A stable finite element for the Stokes equations, Calcolo, Volume 21 (1984) no. 4, p. 337-344 (1985) | DOI | MR | Zbl

[6] Ascher, U. M.; Ruuth, S. J.; Spiteri, R. J. Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., Volume 25 (1997) no. 2–3, pp. 151-167 | DOI | MR | Zbl

[7] Ascher, U. M.; Ruuth, S. J.; Wetton, B. T. R. Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., Volume 32 (1995) no. 3, pp. 797-823 | DOI | MR | Zbl

[8] Bardos, C. W.; Titi, E. S. Mathematics and turbulence: where do we stand?, J. Turbul., Volume 14 (2013) no. 3, pp. 42-76 | MR

[9] Bochev, P. B.; Gunzburger, M. D.; Lehoucq, R. B. On stabilized finite element methods for the Stokes problem in the small time step limit, Internat. J. Numer. Methods Fluids, Volume 53 (2007) no. 4, pp. 573-597 | DOI | MR | Zbl

[10] Boffi, D.; Brezzi, F.; Fortin, M. Mixed Finite Element Methods and Applications, Springer-Verlag Berlin Heidelberg, 2013 | Zbl

[11] Botta, N.; Klein, R.; Langenberg, S.; Lützenkirchen, S. Well balanced finite volume methods for nearly hydrostatic flows, J. Comput. Phys., Volume 196 (2004) no. 2, pp. 539-565 | MR | Zbl

[12] Bowers, A. L.; Cousins, B. R.; Linke, A.; Rebholz, L. G. New connections between finite element formulations of the Navier–Stokes equations, J. Comput. Phys., Volume 229 (2010) no. 24, pp. 9020-9025 | DOI | MR | Zbl

[13] Boyer, F.; Fabrie, P. Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models, Springer-Verlag New York, 2013 | Zbl

[14] Brooks, A.; Hughes, T. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., Volume 32 (1982) no. 1-3, pp. 199-259 FENOMECH ’81, Part I (Stuttgart, 1981) | DOI | MR | Zbl

[15] Buffa, A.; de Falco, C.; Sangalli, G. IsoGeometric Analysis: Stable elements for the 2D Stokes equation, Int. J. Numer. Meth. Fluids, Volume 65 (2011) no. 11–12, pp. 1407-1422 | DOI | MR | Zbl

[16] Burman, E.; Fernández, M. A. Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence, Numer. Math., Volume 107 (2007) no. 1, pp. 39-77 | MR | Zbl

[17] 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

[18] Chorin, A.; Marsden, J. A mathematical introduction to fluid mechanics, Texts in Applied Mathematics, 4, Springer-Verlag, New York, 1993, xii+169 pages | DOI | MR | Zbl

[19] 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

[20] Cotter, C. J.; Shipton, J. Mixed finite elements for numerical weather prediction, J. Comput. Phys., Volume 231 (2012) no. 21, pp. 7076-7091 | DOI | MR | Zbl

[21] Cotter, C. J.; Thuburn, J. A finite element exterior calculus framework for the rotating shallow-water equations, J. Comput. Phys., Volume 257 (2014) no. part B, pp. 1506-1526 | DOI | MR | Zbl

[22] de Frutos, J.; García-Archilla, B.; John, V.; Novo, J. Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements, Adv. Comput. Math., Volume 44 (2018) no. 1, pp. 195-225 | DOI | MR | Zbl

[23] Di Pietro, D.; Ern, A.; Linke, A.; Schieweck, Friedhelm A discontinuous skeletal method for the viscosity-dependent Stokes problem, Comput. Methods Appl. Mech. Engrg., Volume 306 (2016), pp. 175-195 | DOI | MR

[24] Di Pietro, D. Antonio; Ern, A. Mathematical Aspects of Discontinuous Galerkin Methods, Springer-Verlag Berlin, 2012 | Zbl

[25] Dorok, O.; Grambow, W.; Tobiska, L. Aspects of finite element discretizations for solving the Boussinesq approximation of the Navier–Stokes Equations, Notes on Numerical Fluid Mechanics: Numerical Methods for the Navier–Stokes Equations., Volume 47 (1994), pp. 50-61 | Zbl

[26] Drazin, P. G.; Riley, N. The Navier–Stokes Equations: A Classification of Flows and Exact Solutions, Cambridge University Press, 2006 | Zbl

[27] Ern, A.; Guermond, J.-L. Theory and Practice of Finite Elements, Springer New York, 2004 | Zbl

[28] Ethier, C. R.; Steinman, D. A. Exact fully 3D Navier–Stokes solutions for benchmarking, Int. J. Numer. Meth. Fluids, Volume 19 (1994) no. 5, pp. 369-375 | DOI | Zbl

[29] Evans, John A.; Hughes, T. J. R. IsoGeometric divergence-conforming B-splines for the steady Navier–Stokes equations, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 8, pp. 1421-1478 | DOI | MR | Zbl

[30] Evans, John A.; Hughes, T. J. R. Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations, J. Comput. Phys., Volume 241 (2013), pp. 141-167 | DOI | MR | Zbl

[31] Fortin, M.; Fortin, A. Newer and newer elements for incompressible flows, Finite Elements in Fluids, Volume 6 (1985), pp. 171-187 | Zbl

[32] Franca, L.; Hughes, T. Two classes of mixed finite element methods, Comput. Methods Appl. Mech. Engrg., Volume 69 (1988) no. 1, pp. 89-129 | MR | Zbl

[33] Galvin, K.; Linke, A.; Rebholz, L.; Wilson, N. Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Comput. Methods Appl. Mech. Engrg., Volume 237/240 (2012), pp. 166-176 | DOI | MR | Zbl

[34] Ganesan, S.; John, V. Pressure separation — a technique for improving the velocity error in finite element discretisations of the Navier–Stokes equations, Appl. Math. Comp., Volume 165 (2005) no. 2, pp. 275-290 | MR | Zbl

[35] Gerbeau, J.-F.; Le Bris, C.; Bercovier, M. Spurious velocities in the steady flow of an incompressible fluid subjected to external forces, Int. J. Numer. Meth. Fluids, Volume 25 (1997) no. 6, pp. 679-695 | DOI | MR | Zbl

[36] Girault, V.; Nochetto, R. H.; Scott, L. R. Max-norm estimates for Stokes and Navier–Stokes approximations in convex polyhedra, Numer. Math., Volume 131 (2015) no. 4, pp. 771-822 | DOI | MR | Zbl

[37] Gresho, P. M.; Chan, S. T. On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 2: Implementation, Int. J. Numer. Meth. Fluids, Volume 11 (1990) no. 5, pp. 621-659 | DOI | Zbl

[38] Gresho, P. M.; Lee, R. L.; Chan, S. T.; Leone, J. M. A new finite element for incompressible or Boussinesq fluids, Proceedings of the Third International Conference on Finite Elements in Flow Problems, Wiley (1980) | Zbl

[39] Hillewaert, K. Development of the Discontinuous Galerkin Method for High-Resolution, Large Scale CFD and Acoustics in Industrial Geometries, Ph.D. thesis, Université catholique de Louvain, 2013

[40] Jenkins, E.; John, V.; Linke, A.; Rebholz, L. On the parameter choice in grad-div stabilization for the Stokes equations, Adv. Comput. Math., Volume 40 (2014) no. 2, pp. 491-516 | DOI | MR | Zbl

[41] John, V. Finite Element Methods for Incompressible Flow Problems, Springer International Publishing, 2016 | Zbl

[42] 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

[43] Karniadakis, G. E.; Sherwin, S. J. Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press, Oxford, 2005 | DOI | Zbl

[44] Klingenberg, C.; Puppo, G.; Semplice, M. Arbitrary Order Finite Volume Well-Balanced Schemes for the Euler Equations with Gravity, SIAM J. Sci. Comput., Volume 41 (2019) no. 2, p. A695-A721 | DOI | MR | Zbl

[45] Lehrenfeld, C. Hybrid Discontinuous Galerkin methods for solving incompressible flow problems, Rheinisch-Westfälische Technische Hochschule Aachen (2010) (Masters thesis)

[46] Linke, A. A divergence-free velocity reconstruction for incompressible flows, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 17-18, pp. 837-840 | DOI | MR | Zbl

[47] Linke, A. On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Methods Appl. Mech. Engrg., Volume 268 (2014), pp. 782-800 | DOI | MR | Zbl

[48] Linke, A.; Merdon, C. Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., Volume 311 (2016), pp. 304-326 | DOI | MR | Zbl

[49] 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 | MR | Zbl

[50] Linke, A.; Merdon, C.; Neilan, M.; Neumann, F. Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-Problem, Math. Comp., Volume 87 (2018) no. 312, pp. 1543-1566 | DOI | MR | Zbl

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

[52] Linke, Alexander; Matthies, Gunar; Tobiska, Lutz Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors, ESAIM: M2AN, Volume 50 (2016) no. 1, pp. 289-309 | DOI | MR | Zbl

[53] Lube, G.; Olshanskii, M. A. Stable finite-element calculation of incompressible flows using the rotation form of convection, IMA J. Numer. Anal., Volume 22 (2002) no. 3, pp. 437-461 | MR | Zbl

[54] Olshanskii, M. A.; Lube, G.; Heister, T.; Löwe, J. Grad-div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., Volume 198 (2009) no. 49-52, pp. 3975-3988 | DOI | MR | Zbl

[55] Pelletier, D.; Fortin, A.; Camarero, R. Are FEM solutions of incompressible flows really incompressible? (or how simple flows can cause headaches!), Internat. J. Numer. Methods Fluids, Volume 9 (1989) no. 1, pp. 99-112 | DOI | MR | Zbl

[56] Rivière, B. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, SIAM, 2008 | DOI | Zbl

[57] Schieweck, F. Parallele Lösung der Navier–Stokes-Gleichungen, University of Magdeburg (1997) (Habilitation) | Zbl

[58] Schöberl, J. C++11 Implementation of Finite Elements in NGSolve (2014) no. 30/2014 https://www.asc.tuwien.ac.at/~schoeberl/wiki/publications/ngs-cpp11.pdf (ASC Report)

[59] Schroeder, P. W. Robustness of High-Order Divergence-Free Finite Element Methods for Incompressible Computational Fluid Dynamics, Georg-August-Universität Göttingen (2019) https://hdl.handle.net/11858/00-1735-0000-002E-E5BC-8 (Ph. D. Thesis)

[60] Schroeder, P. W.; Lehrenfeld, C.; Linke, A.; Lube, G. Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations, SeMA J., Volume 75 (2018) no. 4, pp. 629-653 | DOI | MR | Zbl

[61] Schroeder, P. W.; Lube, G. Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier–Stokes flows, J. Numer. Math., Volume 25 (2017) no. 4, pp. 249-276 | DOI | MR | Zbl

[62] Thatcher, R. W.; Silvester, D. A locally mass conserving quadratic velocity, linear pressure element, 1987 (Numerical Analysis Report No, 147, Manchester University/UMIST)

[63] Xu, H.; Cantwell, C. D.; Monteserin, C.; Eskilsson, C.; Engsig–Karup, A. P.; Sherwin, S. J. Spectral/hp element methods: Recent developments, applications, and perspectives, J. Hydrodyn., Volume 30 (2018), pp. 1-22 | DOI

[64] Zang, T. A. On the Rotation and Skew-symmetric Forms for Incompressible Flow Simulations, Appl. Numer. Math., Volume 7 (1991) no. 1, pp. 27-40 | Zbl

[65] Zhang, S. A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp., Volume 74 (2005) no. 250, pp. 543-554 | DOI | MR

Cité par Sources :