# ANALYSIS OF VELOCITY-FLUX FIRST-ORDER SYSTEM LEAST-SQUARES PRINCIPLES FOR THE OPTIMAL CONTROL PROBLEMS FOR THE NAVIER-STOKES EQUATIONS

• Accepted : 2010.06.06
• Published : 2010.06.25

#### Abstract

This paper develops a least-squares approach to the solution of the optimal control problem for the Navier-Stokes equations. We recast the optimality system as a first-order system by introducing velocity-flux variables and associated curl and trace equations. We show that a least-squares principle based on $L^2$ norms applied to this system yields optimal discretization error estimates in the $H^1$ norm in each variable.

#### Acknowledgement

Supported by : Korea Research Foundation

#### References

1. P. B. BOCHEV, Analysis of least-squares finite element methods for the Navier-Stokes equations, Siam J. Numer. Anal., 34 (1997), pp. 1817-1844. https://doi.org/10.1137/S0036142994276001
2. P. B. BOCHEV AND M. D. GUNZBURGER, Analysis of least-squares finite element methods for the Stokes equations, Math. Comp., 63 (1994), pp. 479-506. https://doi.org/10.1090/S0025-5718-1994-1257573-4
3. P. B. BOCHEV, Z. CAI, T. A. MANTEUFFEL AND S. F. MCCORMICK, Analysis of velocity-flux leastsquares principles for the Navier-Stokes equations. Part I., SIAM J. Numer. Anal., 35 (1998), pp. 990-1009. https://doi.org/10.1137/S0036142996313592
4. P. B. BOCHEV, T. A. MANTEUFFEL AND S. F. MCCORMICK, Analysis of velocity-flux least-squares principles for the Navier-Stokes equations. Part II., SIAM J. Numer. Anal., 36 (1999), pp. 1125-1144. https://doi.org/10.1137/S0036142997324976
5. J. H. BRAMBLE AND J. E. PASCIAK, Least-squares methods for the Stokes equations on a discrete minus one inner product, Jour. comput. appl. math, 74 (1996), pp. 155-173. https://doi.org/10.1016/0377-0427(96)00022-2
6. Z. CAI, T. MANTEUFFEL, AND S. MCCORMICK, First-order system least squares for the Stokes equations, with application to linear elasticity, SIAM J. Numer. Anal., 34 (1997) pp. 1727-1741. https://doi.org/10.1137/S003614299527299X
7. P. CIARLET, Finite element method for elliptic problems, North-Holland, Amsterdam, 1978.
8. Y. CHOI, S. D. KIM, H.-C. LEE AND B.-C. SHIN, Analysis of first-order system least-squares for the optimal control problems for the Navier-Stokes equations, J. KSIAM, vol.11 No.4, 55-68, 2007
9. Y. CHOI, H.-C. LEE AND S. D. KIM, Analysis and computations of least-squares method for optimal control problems for the Stokes equations, J. Korean Math. Soc. 46 (2009), no. 5, 1007-1025 https://doi.org/10.4134/JKMS.2009.46.5.1007
10. V. GIRAULT AND P.-A. RAVIART, Finite element methods for Navier-Stokes equations, Springer, Berlin, 1986.
11. M. D. GUNZBURGER, L. HOU, AND T. P. SVOBODNY , Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls, Math. Comp., 57 (1991), pp. 123-151. https://doi.org/10.1090/S0025-5718-1991-1079020-5
12. H.-C. LEE AND Y. CHOI, A least-squares method for optimal control problems for a second-order elliptic systems in two dimensions, Jour. Math. Anal. Appl., 242 (2000), pp. 105-128. https://doi.org/10.1006/jmaa.1999.6658
13. SOOROK RYU, HYUNG-CHUN LEE AND SANG DONG KIM, First- order system least-squares methods for an optimal control problem by the Stokes flow, SIAM J. Numer. Anal. 47 (2009), no. 2, 1524-1545. https://doi.org/10.1137/070701157
14. R. TEMAM, Nonlinear Functional Analysis and Navier-Stokes Equations, SIAM, Philadelphia, 1983.