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Analysis of Continuous \(H^{-1}\)-Least-Squares Methods for the Steady Navier–Stokes System

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Abstract

We analyze two \(H^{-1}\)-least-squares methods for the steady Navier–Stokes system of incompressible viscous fluids. Precisely, we show the convergence of minimizing sequences for the least-squares functional toward solutions. Numerical experiments support our analysis.

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References

  1. Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8(1), 141–148 (1988)

    Article  MathSciNet  Google Scholar 

  2. Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods. Applied Mathematical Sciences, vol. 166. Springer, New York (2009)

    MATH  Google Scholar 

  3. Bristeau, M.O., Pironneau, O., Glowinski, R., Periaux, J., Perrier, P.: On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. I. Least square formulations and conjugate gradie. Comput. Methods Appl. Mech. Eng. 17/18(part 3), 619–657 (1979)

    Article  MathSciNet  Google Scholar 

  4. Bristeau, M.O., Pironneau, O., Glowinski, R., Périaux, J., Perrier, P., Poirier, G.: On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. II. Application to transonic flow simulations. Comput. Methods Appl. Mech. Eng. 51(1–3), 363–394 (1985). FENOMECH ’84, Part I, II (Stuttgart, 1984)

    Article  MathSciNet  Google Scholar 

  5. Chehab, J.-P., Raydan, M.: Implicitly preconditioned and globalized residual method for solving steady fluid flows. Electron. Trans. Numer. Anal. 34, 136–151 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Fletcher, R.: On the Barzilai-Borwein Method, Optimization and Control with Applications. Applied Mathematics & Optimization, vol. 96, pp. 235–256. Springer, New York (2005)

    Book  Google Scholar 

  7. Glowinski, R.: Finite element methods for incompressible viscous flow. Numerical Methods for Fluids (Part 3), Handbook of Numerical Analysis, vol. 9, pp. 3–1176. Elsevier, New York (2003)

  8. Glowinski, R.: Variational methods for the numerical solution of nonlinear elliptic problems. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 86. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2015)

  9. Glowinski, R., Mantel, B., Periaux, J., Pironneau, O.: \(H^{-1}\) least squares method for the Navier–Stokes equations, Numerical methods in laminar and turbulent flow. In: Proceedings of the First International Conference, University College of Swansea, Swansea, Halsted, New York-Toronto, pp. 29–42 (1978)

  10. Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2(1), 35–58 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Jiang, B.-N., Povinelli, L.A.: Least-squares finite element method for fluid dynamics. Comput. Methods Appl. Mech. Eng. 81(1), 13–37 (1990)

    Article  MathSciNet  Google Scholar 

  13. Lee, J.D., Sun, Y., Saunders, M.A.: Proximal Newton-type methods for minimizing composite functions. SIAM J. Optim. 24(3), 1420–1443 (2014)

    Article  MathSciNet  Google Scholar 

  14. Lemoine, J., Münch, A.: A continuous least-squares method for the unsteady Navier–Stokes system: analysis and applications (2019)

  15. Lemoine, J., Münch, A.: Resolution of implicit time schemes for the Navier–Stokes system through a least-squares method (2019)

  16. Münch, A.: A least-squares formulation for the approximation of controls for the Stokes system. Math. Control Signals Syst. 27(1), 49–75 (2015)

    Article  MathSciNet  Google Scholar 

  17. Münch, A., Pedregal, P.: A least-squares formulation for the approximation of null controls for the Stokes system. C. R. Math. Acad. Sci. Paris 351(13–14), 545–550 (2013)

    Article  MathSciNet  Google Scholar 

  18. Münch, A., Pedregal, P.: Numerical null controllability of the heat equation through a least squares and variational approach. Eur. J. Appl. Math. 25(3), 277–306 (2014)

    Article  MathSciNet  Google Scholar 

  19. Pedregal, P.: A variational approach for the Navier–Stokes system. J. Math. Fluid Mech. 14(1), 159–176 (2012)

    Article  MathSciNet  Google Scholar 

  20. Pedregal, P.: On error functionals. SeMA J. 65, 13–22 (2014)

    Article  MathSciNet  Google Scholar 

  21. Raydan, M.: On the Barzilai and Borwein choice of steplength for the gradient method. IMA J. Numer. Anal. 13(3), 321–326 (1993)

    Article  MathSciNet  Google Scholar 

  22. Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7(1), 26–33 (1997)

    Article  MathSciNet  Google Scholar 

  23. Temam, R.: Navier–Stokes Equations. AMS Chelsea Publishing, Providence (2001) Theory and numerical analysis, Reprint of the 1984 edition

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Correspondence to Arnaud Münch.

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Research supported by MTM2017-83740-P, by PEII-2014-010-P of the Conserjería de Cultura (JCCM), and by Grant GI20152919 of UCLM.

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Lemoine, J., Münch, A. & Pedregal, P. Analysis of Continuous \(H^{-1}\)-Least-Squares Methods for the Steady Navier–Stokes System. Appl Math Optim 83, 461–488 (2021). https://doi.org/10.1007/s00245-019-09554-5

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