A New Numerical Method for Solving the Stokes Problem Using Quadratic Programming

ABSTRACT

In this paper we present a new method for solving the Stokes problem which is a constrained optimization method. The new method is simpler and requires less computation than the existing methods. In this method we transform the Stokes problem into a quadratic programming problem and by solving it, the velocity and the pressure are obtained.

In this paper we present a new method for solving the Stokes problem which is a constrained optimization method. The new method is simpler and requires less computation than the existing methods. In this method we transform the Stokes problem into a quadratic programming problem and by solving it, the velocity and the pressure are obtained.

Cite this paper

nullM. Baymani and A. Kerayechian, "A New Numerical Method for Solving the Stokes Problem Using Quadratic Programming,"*Intelligent Information Management*, Vol. 2 No. 3, 2010, pp. 199-203. doi: 10.4236/iim.2012.23023.

nullM. Baymani and A. Kerayechian, "A New Numerical Method for Solving the Stokes Problem Using Quadratic Programming,"

References

[1] F. Brezzi and M. Fortin, “Mixed and hybrid finite element methods,” Springer-Verlag, New York, 1991.

[2] H. Elman, D. Silvester, and A. Wathen, “Finite elements and fast iterative solvers with applications in incompressible fluid dynamics,” Oxford University Press, Oxford 2005.

[3] V. Girault, and P. Raviart, “Finite element methods for Navier–Stokes equations,” Springer-Verlag, Berlin, 1986.

[4] T. Barth, P. Bochev, M. Gunzburger, and J. Shahid, “A taxomony of consistently stabilized finite element methods for stokes problem,” SIAM Journal on Scientific Computing, Vol. 25, pp. 1585–1607, 2004.

[5] L. P. Franca, T. J. R. Hughes, and R. Stenberg, “Stabilised finite element methods,” in: Incompressible Computational Fluid Dynamics Trends and Advances, Cambridge University, pp. 87–107, 1993.

[6] N. Kechkar and D. Silvester, “Analysis of locally stabilized mixed finite element methods for Stokes problem,” Mathematics of Computation, Vol. 58, pp. 1–10, 1992.

[7] R. Araya, G. R. Barrenechea, and F. Valentin, “Stabilized finite element methods based on multiscale enrichment for the Stokes problem,” SIAM journal on Numerical Analysis, Vol. 44, No. 1, pp. 322–348, 2006.

[8] G. R. Barrenechea and F. Valentin, “Relationship between multiscale enrichment and stabilized finite element methods for the generalized Stokes problem I,” CR Academic Science, Vol. 341, pp. 635–640, 2005.

[9] E. Burman, M. Fernandez, and P. Hansbo, “Continuous interior penalty finite element method for Oseen’s equations,” SIAM Journal on Numerical Analysis, Vol. 44, No. 6, pp. 1248–1274, 2006.

[10] E. Burman and P. Hansbo, “Edge stabilization for the generalized Stokes problem: A continuous interior penalty method,” Computer Methods on Applied Mechanics and Engneering. Vol. 195, pp. 2393–2410, 2006.

[11] K. Nafa, and A. J. Wathen, “Local projection stabilized Galerkin approximations for the generalized Stokes problem,” Computer Methods on Applied Mechanics and Engneering, Vol. 198, pp. 877–883, 2009.

[12] B. Borre and N. Lukkassen, “Application of homogenization theory related to Stokes flow in porous media,” Applications of Mathematics, Vol. 44, No. 4, pp. 309–319, 1999.

[13] A. Quarteroni and A. Valli, “Numerical approximation of partial differential equations,” ISBN 3–540–57111–6, Springer-Verlag, Berlin Heidelberg, New York, 1997.

[14] S. Effati and M. Baymani, “A new nonlinear neural network for solving quadratic programming problems,” Applied Mathematics and Computational, Vol. 165, pp. 719–729, 2005.

[15] C. C. Tsai and S. Y. Yang, “On the velocity-vorticity- pressure least–squares finite element method for the stationary incompressible Oseen problem,” Journal of Com- putational and Applied Mathematics, Vol. 182, pp. 211–232, 2005.

[1] F. Brezzi and M. Fortin, “Mixed and hybrid finite element methods,” Springer-Verlag, New York, 1991.

[2] H. Elman, D. Silvester, and A. Wathen, “Finite elements and fast iterative solvers with applications in incompressible fluid dynamics,” Oxford University Press, Oxford 2005.

[3] V. Girault, and P. Raviart, “Finite element methods for Navier–Stokes equations,” Springer-Verlag, Berlin, 1986.

[4] T. Barth, P. Bochev, M. Gunzburger, and J. Shahid, “A taxomony of consistently stabilized finite element methods for stokes problem,” SIAM Journal on Scientific Computing, Vol. 25, pp. 1585–1607, 2004.

[5] L. P. Franca, T. J. R. Hughes, and R. Stenberg, “Stabilised finite element methods,” in: Incompressible Computational Fluid Dynamics Trends and Advances, Cambridge University, pp. 87–107, 1993.

[6] N. Kechkar and D. Silvester, “Analysis of locally stabilized mixed finite element methods for Stokes problem,” Mathematics of Computation, Vol. 58, pp. 1–10, 1992.

[7] R. Araya, G. R. Barrenechea, and F. Valentin, “Stabilized finite element methods based on multiscale enrichment for the Stokes problem,” SIAM journal on Numerical Analysis, Vol. 44, No. 1, pp. 322–348, 2006.

[8] G. R. Barrenechea and F. Valentin, “Relationship between multiscale enrichment and stabilized finite element methods for the generalized Stokes problem I,” CR Academic Science, Vol. 341, pp. 635–640, 2005.

[9] E. Burman, M. Fernandez, and P. Hansbo, “Continuous interior penalty finite element method for Oseen’s equations,” SIAM Journal on Numerical Analysis, Vol. 44, No. 6, pp. 1248–1274, 2006.

[10] E. Burman and P. Hansbo, “Edge stabilization for the generalized Stokes problem: A continuous interior penalty method,” Computer Methods on Applied Mechanics and Engneering. Vol. 195, pp. 2393–2410, 2006.

[11] K. Nafa, and A. J. Wathen, “Local projection stabilized Galerkin approximations for the generalized Stokes problem,” Computer Methods on Applied Mechanics and Engneering, Vol. 198, pp. 877–883, 2009.

[12] B. Borre and N. Lukkassen, “Application of homogenization theory related to Stokes flow in porous media,” Applications of Mathematics, Vol. 44, No. 4, pp. 309–319, 1999.

[13] A. Quarteroni and A. Valli, “Numerical approximation of partial differential equations,” ISBN 3–540–57111–6, Springer-Verlag, Berlin Heidelberg, New York, 1997.

[14] S. Effati and M. Baymani, “A new nonlinear neural network for solving quadratic programming problems,” Applied Mathematics and Computational, Vol. 165, pp. 719–729, 2005.

[15] C. C. Tsai and S. Y. Yang, “On the velocity-vorticity- pressure least–squares finite element method for the stationary incompressible Oseen problem,” Journal of Com- putational and Applied Mathematics, Vol. 182, pp. 211–232, 2005.