main-content

Erschienen in:

01.12.2020

H(div) conforming methods for the rotation form of the incompressible fluid equations

verfasst von: Xi Chen, Corina Drapaca

Erschienen in: Calcolo | Ausgabe 4/2020

Einloggen, um Zugang zu erhalten

Abstract

New H(div) conforming finite element methods for incompressible flows are designed that involve the rotation form of the equations of motion and the Bernoulli function. With a specific choice of numerical fluxes, we recover the same velocity field as in Guzmán et al. (IMA J Numer Anal 37(4):1733–1771, 2016) for the incompressible Euler equation in the convection form. Error estimates are presented for the semi-discrete method. We further study the incompressible Navier-Stokes equation with the full version of the stress tensor $$\nu \left( \nabla \varvec{u}+ \nabla \varvec{u}^T - \frac{2}{3} \left( \nabla \cdot \varvec{u}\right) \mathbb {I} \right)$$, instead of partially enforcing the divergence free constraint at the continuous level (as is commonly done in finite element methods), we let the numerical scheme to fully control the enforcement of this constraint. Finally, we test the behavior of the proposed methods with some numerical simulations. Our results show that (1) We recover the same velocity field in Guzmán et al. (2016), (2) When H(div) conforming with BDM-DG elements, we achieve less errors in the velocity compared with Schroeder et al. (SeMA J 75(4):629–653, 2018) when polynomial order $$p\in \{2,3\}$$, (3) When H1 conforming with Taylor-Hood elements, the use of full stress tensor helps to reduce errors in both the velocity and the Bernoulli function, (4) H(div) conforming method does a better job in long time structure preservation compared with the classical mixed method even with the grad-div stabilization.
Fußnoten
1
Unlike other studies which call $$\tilde{p}$$ the Bernoulli pressure, we will use the name of Bernoulli function proposed in [21] because the physical units of the right-hand side of formula (3.2) are not those of a pressure.

Literatur
1.
Ahmed, N.: On the grad-div stabilization for the steady Oseen and Navier-Stokes equations. Calcolo 54(1), 471–501 (2017)
2.
Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21(4), 337–344 (1984)
3.
Bercovier, M., Pironneau, O.: Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33(2), 211–224 (1979)
4.
Boffi, D., Brezzi, F., Fortin, M., et al.: Mixed Finite Element Methods and Applications, vol. 44. Springer, Berlin (2013)
5.
Braack, M., Burman, E., John, V., Lube, G.: Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Eng. 196(4–6), 853–866 (2007)
6.
Burman, E., Linke, A.: Stabilized finite element schemes for incompressible flow using Scott-Vogelius elements. Appl. Numer. Math. 58(11), 1704–1719 (2008)
7.
Chang, C., Nelson, J.J.: Least-squares finite element method for the Stokes problem with zero residual of mass conservation. SIAM J. Numer. Anal. 34(2), 480–489 (1997)
8.
Charnyi, S., Heister, T., Olshanskii, M.A., Rebholz, L.G.: On conservation laws of Navier-Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017)
9.
Chen, X., Williams, D.M.: Versatile mixed methods for the incompressible Navier-Stokes equations. arXiv preprint arXiv:​2007.​08015 (2020)
10.
Clancy, L.J.: Aerodynamics. Halsted Press, London (1975)
11.
Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 74(251), 1067–1095 (2005)
12.
Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69. Springer, Berlin (2011) MATH
13.
Fu, G.: An explicit divergence-free DG method for incompressible flow. Comput. Methods Appl. Mech. Eng. 345, 502–517 (2019)
14.
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. 177(2), 243–267 (2005)
15.
Guzmán, J., Shu, C.W., Sequeira, F.A.: H-(div) conforming and DG methods for incompressible Euler’s equations. IMA J. Numer. Anal. 37(4), 1733–1771 (2016)
16.
Horiuti, K.: Comparison of conservative and rotational forms in large eddy simulation of turbulent channel flow. J. Comput. Phys. 71(2), 343–370 (1987)
17.
Horiuti, K., Itami, T.: Truncation error analysis of the rotational form for the convective terms in the Navier-Stokes equation. J. Comput. Phys. 145(2), 671–692 (1998)
18.
Jenkins, E.W., John, V., Linke, A., Rebholz, L.G.: On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40(2), 491–516 (2014)
19.
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. 59(3), 492–544 (2017)
20.
Kouhia, R., Stenberg, R.: A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng. 124(3), 195–212 (1995)
21.
Kundu, P.K., Cohen, I., Dowling, D.: Fluid Mechanics. 1990. Google Scholar pp. 56–59
22.
Langtangen, H.P., Logg, A.: Solving PDEs in Python. Springer, Berlin (2017). https://​doi.​org/​10.​1007/​978-3-319-52462-7
23.
Layton, W.: Introduction to the numerical analysis of incompressible viscous flows. SIAM (2008)
24.
Layton, W., Manica, C.C., Neda, M., Olshanskii, M., Rebholz, L.G.: On the accuracy of the rotation form in simulations of the Navier-Stokes equations. J. Comput. Phys. 228(9), 3433–3447 (2009)
25.
Layton, W., Manica, C.C., Neda, M., Rebholz, L.G.: Numerical analysis and computational comparisons of the NS-alpha and NS-omega regularizations. Comput. Methods Appl. Mech. Eng. 199(13–16), 916–931 (2010)
26.
Linke, A., Rebholz, L.G.: Pressure-induced locking in mixed methods for time-dependent (Navier-) Stokes equations. arXiv preprint arXiv:​1808.​07028 (2018)
27.
Natale, A., Cotter, C.J.: Scale-selective dissipation in energy-conserving finite-element schemes for two-dimensional turbulence. Quart. J. R. Meteorol. Soc. 143(705), 1734–1745 (2017) CrossRef
28.
Natale, A., Cotter, C.J.: A variational finite-element discretization approach for perfect incompressible fluids. IMA J. Numer. Anal. 38(3), 1388–1419 (2017)
29.
Olshanskii, M., Lube, G., Heister, T., Löwe, J.: Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 198(49–52), 3975–3988 (2009)
30.
Olshanskii, M.A.: A low order Galerkin finite element method for the Navier-Stokes equations of steady incompressible flow: a stabilization issue and iterative methods. Comput. Methods Appl. Mech. Eng. 191(47–48), 5515–5536 (2002)
31.
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. 75(4), 629–653 (2018)
32.
Schroeder, P.W., Lube, G.: Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier-Stokes flows. J. Numer. Math. 5, 69 (2017)
33.
Schroeder, P.W., Lube, G.: Divergence-free H (div)-FEM for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics. J. Sci. Comput. 68, 1–29 (2018)
34.
Zang, T.A.: On the rotation and Skew-symmetric forms for incompressible flow simulations. Appl. Numer. Math. 7(1), 27–40 (1991)
35.
Zhang, S.: The Divergence-Free Finite Elements for the Stationary Stokes Equations. Preprint University of Delaware (2007)
Titel
H(div) conforming methods for the rotation form of the incompressible fluid equations
verfasst von
Xi Chen
Corina Drapaca
Publikationsdatum
01.12.2020
Verlag
Springer International Publishing
Erschienen in
Calcolo / Ausgabe 4/2020
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-020-00380-8

Zur Ausgabe