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A New L2 Projection Method for the Oseen Equations

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  28 November 2017

Yanhong Bai  and
Minfu Feng
Yanhong Bai*
Affiliation:
College of Sciences, and Institute of Nonlinear Dynamics, Southwest Petroleum University, Chengdu, Sichuan 610500, China
Minfu Feng*
Affiliation:
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
*
*Corresponding author. Emails:baiyanhong1982@126.com (Y. H. Bai), fmf@wtjs.cn (M. F. Feng)
*Corresponding author. Emails:baiyanhong1982@126.com (Y. H. Bai), fmf@wtjs.cn (M. F. Feng)
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Abstract

In this paper, a new type of stabilized finite element method is discussed for Oseen equations based on the local L2 projection stabilized technique for the velocity field. Velocity and pressure are approximated by two kinds of mixed finite element spaces, Pl2–P1, (l = 1,2). A main advantage of the proposed method lies in that, all the computations are performed at the same element level, without the need of nested meshes or the projection of the gradient of velocity onto a coarse level. Stability and convergence are proved for two kinds of stabilized schemes. Numerical experiments confirm the theoretical results.

Keywords

Oseen equationsL2 projection methodpressure projection method

MSC classification

Secondary: 65N12: Stability and convergence of numerical methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 6 , December 2017 , pp. 1420 - 1437
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Araya, R., Barrenechea, G. R., Poza, A. H. and Valentin, F., Convergence analysis of a residual local projection finite element method for the Navier-Stokes equations, SIAM J. Numer. Anal., 50 (2012), pp. 669–699.CrossRefGoogle Scholar
[2] Araya, R., Barrenechea, G. R. and Valentin, F., Stabilized finite element methods based on multiscale enrichement for the Stokes problem, SIAM J. Numer. Anal., 44 (2006), pp. 322–348.Google Scholar
[3] Arndt, D., Dallmann, H. and Lube, G., Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem, Numer. Methods Partial Differential Equations, doi: 10.1002/2014/21944.CrossRefGoogle Scholar
[4] Barrenechea, G. R. and Valentin, F., A residual local projection method for the Oseen equation, Computer Methods Appl. Mech. Eng., 199 (2010), pp. 1906–1921.Google Scholar
[5] Barrenechea, G. R. and Valentin, F., Beyond pressure stabilization: A low order local projection method for the Oseen equation, Int. J. Numer. Methods Eng., 86 (2011), pp. 801–815.Google Scholar
[6] Becker, R. and Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo, 38 (2000), pp. 173–199.CrossRefGoogle Scholar
[7] Blasco, J. and Codina, R., Stabilized finite element method for the transient Navier-Stokes equations based on a pressure gradient projection, Computer Methods Appl. Mech. Eng., 182 (2000), pp. 277–300.Google Scholar
[8] Blasco, J. and Codina, R., Space and time error estimates for a first-order, pressure stabilized finite element method for the incompressible Navier-Stokes equations, Appl. Numer. Math., 38 (2001), pp. 475–497.CrossRefGoogle Scholar
[9] Bochev, P., Dohrmann, C. and Gunzburger, M., Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal., 44 (2006), pp. 82–101.CrossRefGoogle Scholar
[10] Bochev, P. and Gunzburger, M. D., Analysis of least squares finite element methods for the Stokes equations, Math. Comput., 63 (1994), pp. 479–506.CrossRefGoogle Scholar
[11] Chen, G. and Feng, M. F., 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
[12] Chen, G., Feng, M. F. and Xie, C. M., A new projection-based stabilized method for steady convection-dominated convection-diffusion equations, Appl. Math. Comput., 239 (2014), pp. 89– 106.Google Scholar
[13] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amesterdam, New York, Oxford, 1978.Google Scholar
[14] Clément, PH., Approximation by finite element functions using local regularization, Math. Model. Numer. Anal., 2 (1975), pp. 77–84.Google Scholar
[15] Codina, R. and Blasco, J., Analysis of a pressure stabilized finite element approximation of the stationary Navier-Stokes equations, Numerische Mathematik, 87 (2000), pp. 59–81.Google Scholar
[16] Dohrmann, C. and Bochev, P., A stabilized finite element method for the Stokes problem based on polynomial pressure projection, Int. J. Numer. Methods Fluids, 46 (2004), pp. 183–201.CrossRefGoogle Scholar
[17] Feng, M. F., Bai, Y. H., He, Y. N. and Qin, Y. M., 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 (2011), pp. 415–440.Google Scholar
[18] Franca, L. P. and Oliveira, S. P., Pressure bubbles stabilization features in the Stokes problem, Compurer Methods Appl. Mech. Eng., 192 (2003), pp. 1929–1937.CrossRefGoogle Scholar
[19] Ge, Z. H., Feng, M. F. and He, Y. N., A stabilized nonconfirming finite element method based on multiscale enrichment for the stationary Navier-Stokes equations, Appl. Math. Comput., 202 (2008), pp. 700–707.Google Scholar
[20] Girault, V. and Raviart, P., Finite Element Methods for Navier-Stokes Equations, Springer, Berlin, 1989.Google Scholar
[21] Hecht, F., New development in FreeFem++, J. Numer. Math., 20 (2012), pp. 251–265.Google Scholar
[22] John, V. and Kaya, S., A finite element variational multiscale method for the Navier-Stokes equations, SIAM J. Sci. Comput., 26 (2005), pp. 1485–1503.CrossRefGoogle Scholar
[23] John, V. and Kaya, S., Finite element error analysis of a variational multiscale method for the Navier-Stokes equations, Adv. Comput. Math., 28 (2008), pp. 43–61.CrossRefGoogle Scholar
[24] John, V., Kaya, S. and Kindl, A., Finite element error analysis for a projection-based variational multiscale method with nonlinear eddy viscosity, J. Math. Anal. Appl., 344 (2008), pp. 627–641.Google Scholar
[25] John, V. and Novo, J., Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations, SIAM J. Numer. Anal., 49 (2011), pp. 1149–1176.CrossRefGoogle Scholar
[26] Kaya, S. and Rivière, B., A two-grid stabilization method for solving the steady-state Navier-Stokes equations, Numer. Methods Partial Differential Equations, 22 (2005), pp. 728–743.Google Scholar
[27] Knobloch, P., A generalization of local projection stabilization for convection-diffusion-reaction equations, SIAM J. Numer. Anal., 48 (2010), pp. 659–680.Google Scholar
[28] Knobloch, P. and Tobiska, L., Improved stability and error analysis for a class of local projection stabilizations applied to the Oseen problem, Numer. Methods Partial Differential Equations, 29 (2013), pp. 206–225.Google Scholar
[29] Li, J., He, Y. N. and Chen, Z. X., A new stabilized finite element method for the transient Navier-Stokes equations, Computer Methods Appl. Mech. Eng., 197 (2007), pp. 22–35.Google Scholar
[30] Matthies, G., Skrzypasz, P. and Toblska, L., A unified convergence analysis for local projection stabilizations applied to the Oseen problem, Math. Model. Numer. Anal., 41 (2007), pp. 713–742.Google Scholar
[31] Scott, L. R. and Zhang, S., Finite element interpolation of nonsmoothing functions satisfying boundary conditions, Math. Comput., 54 (1990), pp. 483–493.CrossRefGoogle Scholar
[32] Zhang, Y. and He, Y. N., A subgrid model for the time-dependent Navier-Stokes equations, Adv. Numer. Anal., doi:10.1155/2009/494829.CrossRefGoogle Scholar