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01.03.2015

An extension of the MAC scheme to locally refined meshes: convergence analysis for the full tensor time-dependent Navier–Stokes equations

verfasst von: Eric Chénier, Robert Eymard, Thierry Gallouët, Raphaèle Herbin

Erschienen in: Calcolo | Ausgabe 1/2015

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Abstract

A variational formulation of the standard marker-and-cell scheme for the approximation of the Navier–Stokes problem yields an extension of the scheme to general 2D and 3D domains and more general meshes. An original discretization of the trilinear form of the nonlinear convection term is proposed; it is designed so as to vanish for discrete divergence free functions. This property allows us to give a mathematical proof of the convergence of the resulting approximate solutions, for the nonlinear Navier–Stokes equations in both steady-state and time-dependent regimes, without any small data condition. Numerical examples (analytical steady and time-dependent ones, inclined driven cavity) confirm the robustness and the accuracy of this method.

Vorheriger Artikel Mimetic finite difference approximation of quasilinear elliptic problems

Nächster Artikel Lower and upper bounds of Stokes eigenvalue problem based on stabilized finite element methods

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Blanc, P.: Convergence of a finite volume scheme on a MAC mesh for the Stokes problem with right-hand side in \(H^{-1}\). In: Finite volumes for complex applications IV, pp. 133–142. ISTE, London (2005)

Bramble, J.H.: A proof of the inf-sup condition for the Stokes equations on Lipschitz domains. Math. Models Meth. Appl. Sci. 13(3), 361–371 (2003). (Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday)

Chénier, R., Eymard, R., Herbin, R.: An Extension of the Mac Scheme to Some Unstructured Meshes. In: Finite Volumes for Complex Applications VI, vol. 1, pp. 253–261. Springer, London, Finite Volumes for Complex Applications VI (FVCA VI). Czech Republic, June, Prague (2011)

Chou, S.H., Kwak, D.Y.: A covolume method based on rotated bilinears for the generalized Stokes problem. SIAM J. Numer. Anal. 35(2), 494–507 (1998). (electronic)CrossRefMATHMathSciNet

Demirdzic, I., Lilek, Z., Peric, M.: Fluid flow and heat transfer test problems for non-orthogonal grids: Bench-mark solutions. Int. J. Numer. Meth. Fluids 15, 329–354 (1992)CrossRef

Eymard, R., Fuhrmann, J., Linke, A.: MAC Schemes on Triangular Meshes. In: Finite Volumes for Complex Applications VI, vol. 1, pp. 399–407. Springer, London. Finite Volumes for Complex Applications VI (FVCA VI), Czech Republic, June, Prague (2011)

Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.-L. (eds.) Techniques of Scientific Computing. Part III, Handbook of Numerical Analysis, VII, pp. 713–1020. North-Holland, Amsterdam (2000)

Eymard, R., Gallouët, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30(4), 1009–1043 (2010)CrossRefMATHMathSciNet

Eymard, R., Herbin, R., Latché, J.-C.: Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3d meshes. SIAM J. Numer. Anal. 45(1), 1–36 (2007)CrossRefMATHMathSciNet

10.

Fabrie, P., Gallouët, T.: Modeling wells in porous media flow. Math. Models Meth. Appl. Sci. 10(5), 673–709 (2000)CrossRefMATH

11.

Gallouët, T., Latché, J.-C.: Compactness of discrete approximate solutions to parabolic pdes, application to a turbulence model. Commun. Pure Appl. Anal. 11(6), 2371–2391 (2012)CrossRefMATHMathSciNet

12.

Girault, V., Lopez, H.: Finite-element error estimates for the MAC scheme. IMA J. Numer. Anal. 16(3), 247–379 (1996)CrossRefMathSciNet

13.

Han, H., Wu, X.: A new mixed finite element formulation and the MAC method for the Stokes equations. SIAM J. Numer. Anal. 35(2), 560–571 (1998). (electronic)CrossRefMATHMathSciNet

14.

Harlow, F., Welch, J.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids 8, 2182–2189 (1965)CrossRefMATH

15.

Kanschat, G.: Divergence-free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme. Internat. J. Numer. Meth. Fluids 56(7), 941–950 (2008)CrossRefMATHMathSciNet

16.

Nečas, J.: Les méthodes Directes en théorie des équations Elliptiques. Masson et Cie, Éditeurs (1967)MATH

17.

Nicolaïdes, R.: Analysis and convergence of the mac scheme i: the linear problem. SIAM J. Numer. Anal. 29, 1579–1591 (1992)CrossRefMATHMathSciNet

18.

Nicolaïdes, R., Wu, X.: Analysis and convergence of the mac scheme ii. Navier-Stokes equations. Math. Comp. 65, 29–44 (1996)CrossRefMATHMathSciNet

19.

Nicolaides, R.A.: The Covolume Approach to Computing Incompressible Flows. In: Incompressible Computational Fluid Dynamics: Trends and Advances, pp. 295–333. Cambridge University Press, Cambridge (2008)

20.

Patankar, S.: Numerical Heat Transfer and Fluid Flow. Series in Computational Methods in Mechanics and Thermal Sciences, vol. XIII. Hemisphere Publishing Corporation, New York. McGraw-Hill Book Company, New York (1980)

21.

Porsching, T.A.: Error estimates for MAC-like approximations to the linear Navier-Stokes equations. Numer. Math. 29(3), 291–306 (1978)CrossRefMATHMathSciNet

22.

Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990)CrossRefMATHMathSciNet

23.

Simon, J.: Compact sets in the space lp(0, t;b). Annali di Matematica Pura ed Applicata 146, 65–96 (1987)CrossRefMATH

24.

Temam, R.: Navier-Stokes Equations, volume 2 of Studies in Mathematics and its Applications, 3rd (edn.). Theory and Numerical Analysis, with an Appendix by F, Thomasset. North-Holland Publishing Co., Amsterdam (1984)

25.

Vidović, D., Segal, A., Wesseling, P.: A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids. J. Comput. Phys. 217(2), 277–294 (2006)CrossRefMATHMathSciNet

26.

Wesseling, P.: Principles of Computational Fluid Dynamics. Springer, New York (2001)CrossRef

Titel: An extension of the MAC scheme to locally refined meshes: convergence analysis for the full tensor time-dependent Navier–Stokes equations
verfasst von: Eric Chénier
Robert Eymard
Thierry Gallouët
Raphaèle Herbin
Publikationsdatum: 01.03.2015
Verlag: Springer Milan
Erschienen in: Calcolo / Ausgabe 1/2015
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-014-0108-x

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