Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-22T09:17:44.542Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of Solving Galerkin Finite Element Methods with Symmetric Pressure Stabilization for the Unsteady Navier-Stokes Equations Using Conforming Equal Order Interpolation

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  09 January 2017

Gang Chen  and
Minfu Feng
Gang Chen
Affiliation:
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Minfu Feng*
Affiliation:
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
*
*Corresponding author. Email:fmf@wtjs.cn (M. F. Feng)
Get access

Abstract

This paper gives analysis of a semi-discrete scheme using equal order interpolation to solve unsteady Navier-Stokes equations. A unified pressure stabilized term is added to our scheme. We proved the uniform error estimates with respect to the Reynolds number, provided the exact solution is smooth.

Keywords

Unsteady Navier-Stokes equationssymmetric pressure stabilizationequal order interpolation

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 2 , April 2017 , pp. 362 - 377
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Brooks, A. N. and Hughes, T. J. R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incom-pressible Navier-Stokes equations, Comput. Meth. Appl. Mech. Eng., 32(1-3), pp. 199–259.Google Scholar
[2] Johnson, C. N. and Pitkäranta, U. J., Finite element methods for linear hyperbolic problems, Comput. Meth. Appl. Mech. Eng., 45(1-3) (1984), pp. 285–312.CrossRefGoogle Scholar
[3] Johnson, C. and Saranen, J., Streamline diffusion methods for the incompressible Euler and Navier-Stokes equation, Math. Comput., 47(175) (1986), pp. 1–18.CrossRefGoogle Scholar
[4] Hansbo, P. and Szepessy, A., A velocity pressure streamline diffusion finite-element method for the incompressible Navier-Stokes equations, Comput. Meth. Appl. Mech. Eng., 84(2) (1990), pp. 175–192.CrossRefGoogle Scholar
[5] Zhou, T. and Feng, M., A least squares Petrov-Galerkin finite element method for the stationary Navier-Stokes equations, Math. Comput., 60(202) (1993), pp. 531–543.Google Scholar
[6] Sun, C. and Shen, H., The finite difference streamline diffusion methods for time-dependent convection-diffusion equations, Numer. Math., Chinese University, English Series, 7(1) (1998), pp. 72–85.Google Scholar
[7] Zhang, Q. and Sun, C., Finite difference-streamline diffusion method for nonlinear convection-diffusion equation, Math. Numer. Sinica, 20(2) (1998), pp. 211–224.Google Scholar
[8] Sun, T. and Ma, K., The finite difference streamline diffusion methods for the incompressible Navier-Stokes equations, Appl. Math. Comput., 149(2) (2004), pp. 493–505.Google Scholar
[9] Zhang, Q., Finite difference streamline diffusion method for incompressible N-S equations, Math. Numer. Sinica, 25(3) (2003), pp. 311–320.Google Scholar
[10] Chen, G., Feng, F. and He, Y., Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations, Appl. Math. Mech., 34(9) (2013), pp. 1083–1096.CrossRefGoogle Scholar
[11] Layton, W., A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput., 133 (2002), pp. 147–157.Google Scholar
[12] John, V. and Kaya, S., Finite element error analysis of a variational multiscale method for the Navier-Stokes equaions, Adv. Comput. Math., 28 (2008), pp. 43–61.CrossRefGoogle Scholar
[13] Belenli, M. A., Kaya, S. and Rebholz, L. G., A subgrid stabilization finite element method for incompressible magnetohydrodynamics, Int. J. Comput. Math., 90(7) (2013), pp. 1506–1523.CrossRefGoogle Scholar
[14] Feng, M., Bai, Y., He, Y. and Qin, Y., A new stabilized subgrid eddy viscosity method based on pressure projection and extrapolated trapezoidal rule for the transient Navier-Stokes equations, J. Comput. Math., 29(4) (2011), pp. 415–440.Google Scholar
[15] Wen, J., Feng, M. and He, Y., Convergence analysis of a new multiscale finite element method with the P1/P0 element for the incompressible flow, Comput. Meth. Appl. Mech., Eng., 258(1) (2013), pp. 13–25.Google Scholar
[16] Codina, R., Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales, Appl. Numer. Math., 58 (2008), pp. 264–283.Google Scholar
[17] Burman, E. and Fernandez, M. A., Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence, Numer. Math., 107 (2007), pp. 39–77.Google Scholar
[18] Braack, M. and Burman, E., Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method, SIAM J. Numer. Anal., 43(6) (2006), pp. 2544–2566.CrossRefGoogle Scholar
[19] Matthies, G., Skrzypacz, P. and Tobiska, L., A unified convergence analysis for local projection stablisations applied to the Oseen problem, Math. Model. Numer. Anal., 41(4) (2007), pp. 713–742.CrossRefGoogle Scholar
[20] Becker, R. and Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo, 38 (2001), pp. 173–199.Google Scholar
[21] Chen, G. and Feng, M., A new absolutely stable simplified Galerkin Least-Squares finite element method using nonconforming element for the Stokes problem, Appl. Math. Comput., 219 (2013), pp. 5356–5366.Google Scholar
[22] Chen, G., Feng, M. and He, Y., Unified analysis for stabilized methods of low-order mixed finite elements for stationary Navier-Stokes eqautions, Appl. Math. Mech., 34(8) (2013), pp. 953–970.Google Scholar
[23] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, 1978.Google Scholar
[24] Hecht, F., New development in Freefem++, J. Numer. Math., 20(3-4) (2012), pp. 251–265.CrossRefGoogle Scholar
[25] Chorin, S. J., Numerical solution for the Navier-Stokes equations, Math. Comput., 22 (1968), pp. 745–762.CrossRefGoogle Scholar
[26] Tafti, D., Comparison of some upwind-biased high-order formulations with a second order central-difference scheme for time integration of the incompressible Navier-Stokes equations, Comput. Fluids, 25 (1996), pp. 647–665.Google Scholar
[27] John, V. and Layton, W., Analysis of numerical errors in large eddy simulation, SIAM J. Numer. Anal., 40 (2002), pp. 995–1020.CrossRefGoogle Scholar
[28] Shafer, M. and Turek, S., Benchmark computations of laminar flow around cylinder, Flow Simulation with High-Performance Computers II, Vieweg, 1996.Google Scholar
[29] John, V., Reference values for drag and lift of a two-dimensional time-dependent flow around the cylinder, Int. J. Numer. Methods Fluids, 44 (2004), pp. 777–788.CrossRefGoogle Scholar
[30] Layton, W., Rebholz, L. G. and Trenchea, C., Modular nonlinear filter stabilization of methods for higher Reynolds numbers flow, J. Math. Fluid Mech., 14 (2012), pp. 325–354.Google Scholar
[31] Shi, Z. and Wang, M., Finite Element Methods, Vol. 58 of Series in Information and Computation Science, Science Press, Beijing, 2013.Google Scholar
[32] Li, J. and He, Y. N., A stabilized finite element method based on two local Gauss integrations for the Stokes equations, J. Comput. Appl. Math., 214(1) (2008), pp. 58–65.Google Scholar
[33] He, Y. N. and Li, J., A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations, Appl. Numer. Math., 58(10) (2008), pp. 1503–1514.Google Scholar
[34] Li, J., He, Y. N. and Chen, Z. X., A new stabilized finite element method for the transient Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 197(1-4) (2007), pp. 22–35.CrossRefGoogle Scholar
[35] Li, J., He, Y. N. and Xu, H., A multi-level stabilized finite element method for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 196(29-30) (2007), pp. 2852–2862.Google Scholar
[36] Ganesan, S., Matthies, G. and Tobiska, L., Local projection stabilization of equal order interpolation applied to the Stokes problem, Math. Comput., 77(264) (2008), pp. 2039–2060.Google Scholar