Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CONVERGENCE OF THE NEWTON'S METHOD FOR AN OPTIMAL CONTROL PROBLEMS FOR NAVIER-STOKES EQUATIONS

  • Received : 2010.04.06
  • Published : 2011.09.30
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Abstract

We consider the Newton's method for an direct solver of the optimal control problems of the Navier-Stokes equations. We show that the finite element solutions of the optimal control problem for Stoke equations may be chosen as the initial guess for the quadratic convergence of Newton's algorithm applied to the optimal control problem for the Navier-Stokes equations provided there are sufficiently small mesh size h and the moderate Reynold's number.

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References

  1. P. B. Bochev, Z. Cai, T. A. Manteuffel, and S. F. McCormick, Analysis of velocity-ux first-order system least-squares principles for the Navier-Stokes equations. I, SIAM J. Numer. Anal. 35 (1998), no. 3, 990-1009. https://doi.org/10.1137/S0036142996313592
  2. F. Brezzi, J. Rappaz, and P. A. Raviart, Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions, Numer. Math. 36 (1980/81), no. 1, 1-25.
  3. P. Ciarlet, Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
  4. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer, Berlin, 1986.
  5. S. D. Kim, Y. H. Lee, and B. C. Shin, Newton's method for the Navier-Stokes equations with finite-element initial guess of Stokes equations, Comput. Math. Appl. 51 (2006), no. 5, 805-816. https://doi.org/10.1016/j.camwa.2006.03.007
  6. D. A. Knoll and V. A. Mousseau, On Newton-Krylov multigrid methods for the incom- pressible Navier-Stokes equations, J. Comput. Phys. 163 (2000), 262-267. https://doi.org/10.1006/jcph.2000.6561
  7. D. A. Knoll and W. Rider, A multigrid preconditioned Newton-Krylov method, SIAM J. Sci. Comput. 21 (1999), no. 2, 691-710. https://doi.org/10.1137/S1064827598332709
  8. M. Pernice and M. D. Tocci, A multigrid-preconditioned Newton-Krylov Method for the incompressible Navier-Stokes equations, SIAM J. Sci. Comput. 23 (2001), no. 2, 398-418. https://doi.org/10.1137/S1064827500372250