A Finite Element Approximation of the Stokes Equations

ABSTRACT

In this work, a numerical solution of the incompressible Stokes equations is proposed. The method suggested is based on an algorithm of discretization by the unstable of Q_{1} – P_{0} velocity/pressure ?nite element approximation. It is shown that the inf-sup stability constant is O(h) in two dimensions and O( h_{2}) in three dimensions. The basic tool in the analysis is the method of modi?ed equations which is applied to ?nite difference representations of the underlying ?nite element equations. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

In this work, a numerical solution of the incompressible Stokes equations is proposed. The method suggested is based on an algorithm of discretization by the unstable of Q

Cite this paper

nullS. Alami, A. Elakkad, J. El-Mekaoui, A. Elkhalfi and M. Hammoumi, "A Finite Element Approximation of the Stokes Equations,"*World Journal of Mechanics*, Vol. 1 No. 6, 2011, pp. 299-305. doi: 10.4236/wjm.2011.16038.

nullS. Alami, A. Elakkad, J. El-Mekaoui, A. Elkhalfi and M. Hammoumi, "A Finite Element Approximation of the Stokes Equations,"

References

[1] F. Brezzi and M. Fortin, “Mixed and Hybrid Finite Element Methods,” Springer-Verlag, New York, 1991. doi:10.1007/978-1-4612-3172-1

[2] J. Douglas and J. Wang, “An Absolutely Stabilized Finite Element Method for the Stokes Problem,” Mathematics of Computation, Vol. 52, No. 186, 1989, pp. 495-508. doi:10.1090/S0025-5718-1989-0958871-X

[3] L. Franca and T. Hughes, “Convergence Analyses of Ga- lerkin Least-Squares Methods for Symmetric Advecti- vediffusive Forms of the Stokes and Incompressible Navier-Stokes Equations,” Journal of Computer Methods in Applied Mechanics and Engineering, Vol. 105, No. 2, 1993, pp. 285-298. doi:10.1016/0045-7825(93)90126-I

[4] T. Hughes and L. Franca, “A New Finite Element Formulation for Computational Fluid Dynamics: VII. The Stokes Problem with Various Well-Posed Boundary Con- ditions: Symmetric Formulations That Converge for All Velocity/Pressure Spaces,” Journal of Computer Methods in Applied Mechanics and Engineering, Vol. 65, No. 1, 1987, pp. 85-96. doi:10.1016/0045-7825(87)90184-8

[5] J. Boland and R. Nicolaides, “On the Stability of Bilinear-Constant Velocity-Pressure Finite Elements,” Journal of Numerical Mathematics, Vol. 44, No. 2, 1984, pp. 219- 222. doi:10.1007/BF01410106

[6] J. Pitkaranta and R. Stenberg, “Error Bounds for the Approximation of the Stokes Problem Using Bilinear/Con- stant Elements on Irregular Quadrilateral Meshes,” In: J. Whiteman, Ed., The Mathematics of Finite Elements and Applications V, Academic Press, London, 1985, pp. 325- 334.

[7] D. Malkus, “Eigenproblems Associated with the Discrete LBB Condition for Incompressible Finite Elements,” In- ternational Journal of Engineering Science, Vol. 19, No. 10, 1981, pp. 1299-1310.

[8] R. Sani, P. Gresho, R. Lee and D. Gri?ths, “The Cause and Cure (?) of the Spurious Pressures Generated by Certain Finite Element Method Solutions of the Incompres- sible Navier-Stokes Equations,” International Journal for Numerical Methods in Fluids, Vol. 1, No. 1, 1981, pp. 17-43. doi:10.1002/fld.1650010104

[9] J. Kevorkian and J. Cole, “Perturbation Methods in Applied Mathematics,” Springer-Verlag, New York, 1981.

[10] A. Elakkad, A. Elkhal? and N. Guessous, “A Posteriori Error Estimation for Stokes Equations,” Journal of Ad- vanced Research in Applied Mathematics, Vol. 2, No. 1, 2010, pp. 1-16. doi:10.5373/jaram.222.092109

[11] E. Erturk, T. C. Corke and C. Gokcol, “Numerical Solutions of 2-D Steady Incompressible Driven Cavity Flow at High Reynolds Numbers,” International Journal for Numerical Methods in Fluids, Vol. 48, No. 7, 2005, pp. 747-774. doi:10.1002/fld.953

[12] U. Ghia, K. Ghia and C. Shin, “High-Re Solution for In- compressible Viscous Flow Using the Navier-Stokes Equa- tions and a Multigrid Method,” Journal of Computational Physics, Vol. 48, No. 3, 1982, pp. 387-395. doi:10.1016/0021-9991(82)90058-4

[13] S. Garcia, “The Lid-Driven Square Cavity Flow: From Stationary to Time Periodic and Chaotic,” Journal of Communications in Computational Physics, Vol. 2, 2007, pp. 900-932.

[1] F. Brezzi and M. Fortin, “Mixed and Hybrid Finite Element Methods,” Springer-Verlag, New York, 1991. doi:10.1007/978-1-4612-3172-1

[2] J. Douglas and J. Wang, “An Absolutely Stabilized Finite Element Method for the Stokes Problem,” Mathematics of Computation, Vol. 52, No. 186, 1989, pp. 495-508. doi:10.1090/S0025-5718-1989-0958871-X

[3] L. Franca and T. Hughes, “Convergence Analyses of Ga- lerkin Least-Squares Methods for Symmetric Advecti- vediffusive Forms of the Stokes and Incompressible Navier-Stokes Equations,” Journal of Computer Methods in Applied Mechanics and Engineering, Vol. 105, No. 2, 1993, pp. 285-298. doi:10.1016/0045-7825(93)90126-I

[4] T. Hughes and L. Franca, “A New Finite Element Formulation for Computational Fluid Dynamics: VII. The Stokes Problem with Various Well-Posed Boundary Con- ditions: Symmetric Formulations That Converge for All Velocity/Pressure Spaces,” Journal of Computer Methods in Applied Mechanics and Engineering, Vol. 65, No. 1, 1987, pp. 85-96. doi:10.1016/0045-7825(87)90184-8

[5] J. Boland and R. Nicolaides, “On the Stability of Bilinear-Constant Velocity-Pressure Finite Elements,” Journal of Numerical Mathematics, Vol. 44, No. 2, 1984, pp. 219- 222. doi:10.1007/BF01410106

[6] J. Pitkaranta and R. Stenberg, “Error Bounds for the Approximation of the Stokes Problem Using Bilinear/Con- stant Elements on Irregular Quadrilateral Meshes,” In: J. Whiteman, Ed., The Mathematics of Finite Elements and Applications V, Academic Press, London, 1985, pp. 325- 334.

[7] D. Malkus, “Eigenproblems Associated with the Discrete LBB Condition for Incompressible Finite Elements,” In- ternational Journal of Engineering Science, Vol. 19, No. 10, 1981, pp. 1299-1310.

[8] R. Sani, P. Gresho, R. Lee and D. Gri?ths, “The Cause and Cure (?) of the Spurious Pressures Generated by Certain Finite Element Method Solutions of the Incompres- sible Navier-Stokes Equations,” International Journal for Numerical Methods in Fluids, Vol. 1, No. 1, 1981, pp. 17-43. doi:10.1002/fld.1650010104

[9] J. Kevorkian and J. Cole, “Perturbation Methods in Applied Mathematics,” Springer-Verlag, New York, 1981.

[10] A. Elakkad, A. Elkhal? and N. Guessous, “A Posteriori Error Estimation for Stokes Equations,” Journal of Ad- vanced Research in Applied Mathematics, Vol. 2, No. 1, 2010, pp. 1-16. doi:10.5373/jaram.222.092109

[11] E. Erturk, T. C. Corke and C. Gokcol, “Numerical Solutions of 2-D Steady Incompressible Driven Cavity Flow at High Reynolds Numbers,” International Journal for Numerical Methods in Fluids, Vol. 48, No. 7, 2005, pp. 747-774. doi:10.1002/fld.953

[12] U. Ghia, K. Ghia and C. Shin, “High-Re Solution for In- compressible Viscous Flow Using the Navier-Stokes Equa- tions and a Multigrid Method,” Journal of Computational Physics, Vol. 48, No. 3, 1982, pp. 387-395. doi:10.1016/0021-9991(82)90058-4

[13] S. Garcia, “The Lid-Driven Square Cavity Flow: From Stationary to Time Periodic and Chaotic,” Journal of Communications in Computational Physics, Vol. 2, 2007, pp. 900-932.