Hostname: page-component-5c6d5d7d68-7tdvq Total loading time: 0 Render date: 2024-08-17T05:02:00.006Z Has data issue: false hasContentIssue false
  • English
  • Français

A Stabilized Finite Element Method for Non-Stationary Conduction-Convection Problems

Published online by Cambridge University Press:  03 June 2015

Ke Zhao ,
Yinnian He  and
Tong Zhang
Ke Zhao*
Affiliation:
Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, Shanxi, China
Yinnian He*
Affiliation:
Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, Shanxi, China
Tong Zhang*
Affiliation:
School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo 454003, Henan, China
*
Email: mathszk@163.com
Corresponding author. URL: http://gr.xjtu.edu.cn:8080/web/heyn Email: heyn@mail.xjtu.edu.cn
Email: zhangtong0616@163.com
Get access

Abstract

This paper is concerned with a stabilized finite element method based on two local Gauss integrations for the two-dimensional non-stationary conduction-convection equations by using the lowest equal-order pairs of finite elements. This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf-sup condition. The stability of the discrete scheme is derived under some regularity assumptions. Optimal error estimates are obtained by applying the standard Galerkin techniques. Finally, the numerical illustrations agree completely with the theoretical expectations.

Keywords

65N3065N1265Z05Non-stationary conduction-convection equationsfinite element methodstabilized methodstability analysiserror estimate
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 2 , April 2011 , pp. 239 - 258
Copyright
Copyright © Global-Science Press 2011

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] 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
[2] 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.Google Scholar
[3] Bochev, P. and Gunzburger, M., An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations, SIAM J. Numer. Anal., 42 (2004), pp. 1189–1207.Google Scholar
[4] Chen, Z. X., Finite Element Methods and Their Applications, Spring-Verlag, Heidelberg, 2005.Google Scholar
[5] Christon, M. A., Gresho, P. M. and Sutton, S. B., Computational predictability of time-dependent natural convection flows in enclosures (including a benchmark solution), Int. J. Numer. Methods. Fluids., (Special Issue: MIT Special Issue on Thermal Convection), 40 (2002), pp. 953–980.Google Scholar
[6] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[7] Codina, R. and Blasco, J., A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation, Comput. Meth. Appl. Mech. Eng., 143 (1997), pp. 373–391.Google Scholar
[8] Codina, R. and Blasco, J., Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations, Numer. Math., 87 (2000), pp. 59–81.Google Scholar
[9] Codina, R., Blasco, J., Buscaglia, G. and Huerta, A., Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection, Int. J. Numer. Methods. Fluids., 37 (2001), pp. 419–444.Google Scholar
[10] Dibenedetto, E. and Friedman, A., Conduction-convection problems with change of phase, J. Differ. Equ., 62 (1986), pp. 129–185.Google Scholar
[11] Douglas, J. J. and Wang, J. P., An absolutely stabilized finite element method for the Stokes problem, Math. Comput., 52 (1989), pp. 495–508.Google Scholar
[12] Franca, L. and Frey, F., Stabilized finite element methods: II, the incompressible Navier-Stokes equations, Comput. Meth. Appl. Mech. Eng., 99 (1992), pp. 209–233.Google Scholar
[13] Franca, L. and Hughes, T., Convergence analyses of Galerkin-least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations, Com-put. Meth. Appl. Mech. Eng., 105 (1993), pp. 285–298.Google Scholar
[14] Franca, L. and Stenberg, R., Error analysis of some Galerkin least squares methods for the elasticity equations, SIAM J. Numer. Anal., 28 (1991), pp. 1680–1697.Google Scholar
[15] Girault, V. and Raviart, P. A., Finite Element Method for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin and Heidelberg, 1987.Google Scholar
[16] Gresho, P. M., Lee, R. L. and Chen, S. T., Solution of the time-dependent incompressible Navier-Stokes and Boussinesq equation using the Galerkin finite element method, Approximation Methods for the Navier-Stokes Problems, Springer Lecture Notes in Mathematics, Springer, Berlin, 1980, pp. 203–222.Google Scholar
[17] Heywood, J. G. and Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem I: regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), pp. 275–311.Google Scholar
[18] Hughes, T., Franca, L. and Balestra, M., A new finite element formulation for computational fluid dynamics: V, circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Meth. Appl. Mech. Eng., 59 (1986), pp. 85–99.Google Scholar
[19] Kay, D. and Silvester, D., A posteriori error estimation for stabilized mixed approximations of the Stokes equations, SIAM J. Sci. Comput., 21 (2000), pp. 1321–1336.Google Scholar
[20] 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 (2008), pp. 58–65.Google Scholar
[21] Li, J., He, Y. N. and Chen, Z. X., A new stabilized finite element method for the transient Navier-Stokes equations, Comput. Meth. Appl. Mech. Eng., 197 (2007), pp. 22–35.Google Scholar
[22] Luo, Z. D., Chen, J., Navon, I. and Zhu, J., An optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems, Int. J. Numer. Meth. Fluids., 60 (2009), pp. 409–436.Google Scholar
[23] Mesquita, M. and Lemos, M., Optimal multigrid solutions of two-dimensional convection-conduction problems, Appl. Math. Comput., 152 (2004), pp. 725–742.Google Scholar
[24] Nader, B. C., Brahim, B. B. and Taieb, L., A multigrid method for solving the Navier-Stokes/Boussinesq equations, Commun. Numer. Meth. Engng., 24 (2008), pp. 671–681.Google Scholar
[25] Shen, J., On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM J. Numer. Anal., 32 (1995), pp. 386–403.Google Scholar
[26] Si, Z. Y. and He, Y. N., A coupled Newton iterative mixed finite element method for stationary conduction-convection problems, Comput., 89 (2010), pp. 1–25.Google Scholar
[27] Si, Z. Y., Zhang, T. and Wang, K., A Newton iterative mixed finite element method for stationary conduction-convection problems, Int. J. Comput. Fluid. Dyn., 24 (2010), pp. 135–141.Google Scholar
[28] Temam, R., Navier-Stokes Equation: Theory and Numerical Analysis (Third edition), North-Holland, Amsterdam, New York, Oxford, 1984.Google Scholar