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Error Analysis of a Unconditionally Stable BDF2 Finite Element Scheme for the Incompressible Flows with Variable Density

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Abstract

Based on an equivalent form of the variable density flows, we propose and study a second-order linearized finite element scheme for the approximation of the three-dimensional incompressible Navier–Stokes equations with variable density, where the two-step backward differentiation formula is used in the discretization of time derivative. It is shown that the proposed finite element scheme is unconditionally stable in the sense that the discrete energy inequalities hold without any condition on the time step size and mesh size. By a rigorous error analysis, the optimal second-order convergence rate is proved in \(L^2\)-norm. Finally, numerical results are provided to confirm our theoretical analysis.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H., Welcome, M.L.: A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys. 142, 1–46 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. An, R.: Error analysis of a time-splitting method for incompressible flows with variable density. Appl. Numer. Math. 150, 384–395 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. An, R.: Error analysis of a new fractional-step method for the incompressible Navier-Stokes equations with variable density. J. Sci. Comput. 84, 3 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. An, R.: Iteration penalty method for the incompressible Navier-Stokes equations with variable density based on the artificial compressible method. Adv. Comput. Math. 46, 5 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bell, J.B., Marcus, D.L.: A second-order projection method for variable-density flows. J. Comput. Phys. 101, 334–348 (1992)

    Article  MATH  Google Scholar 

  6. Blasco, J., Codina, R.: Error estimates for an operator-splitting method for incompressible flows. Appl. Numer. Math. 51, 1–17 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  8. Cai, W.T., Li, B.Y., Li, Y.: Error analysis of a fully discrete finite element method for variable density incompressible flows in two dimensions. ESAIM: M2AN 55, S103–S147 (2021)

    Article  MathSciNet  Google Scholar 

  9. Chen, H.T., Mao, J.J., Shen, J.: Error estimate of Gauge-Uzawa methods for incompressible flows with variable density. J. Comput. Appl. Math. 364, 112321 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comput. 22, 745–762 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  11. Weinan, E., Liu, J.G.: Gauge method for viscous incompressible flows. Commun. Math. Sci. 1, 317–332 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gerbeau, J., Le Bris, C.: Existence of solution for a density-dependent magnetohydrodynamic equation. Adv. Differ. Equ. 2, 427–452 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Guermond, J.L., Quartapelle, L.: A projection FEM for variable density incompressible flows. J. Comput. Phys. 165, 167–188 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guermond, J.L., Salgado, A.: A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228, 2834–2846 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guermond, J.L., Salgado, A.: Error analysis of a fractional time-stepping technique for incompressible flows with variable density. SIAM J. Numer. Anal. 49, 917–944 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Heywood, J., Rannacher, R.: Finite-element approximation of the nonstationary Navier–Stokes problem Part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, B.Y., Qiu, W.F., Yang, Z.Z.: A convergent post-processed discontinuous Galerkin method for incompressible flow with variable density. J. Sci. Comput. 91, 2 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, M.J., Cheng, Y.P., Shen, J., Zhang, X.X.: A bound-preserving high order scheme for variable density incompressible Navier–Stokes equations. J. Comput. Phys. 425, 109906 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, Y., Li, J., Mei, L.Q., Li, Y.P.: Mixed stabilized finite element methods based on backward difference/Adams-Bashforth scheme for the time-dependent variable density incompressible flows. Comput. Math. Appl. 70, 2575–2588 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, Y., Mei, L.Q., Ge, J.T., Shi, F.: A new fractional time-stepping method for variable density incompressible flows. J. Comput. Phys 242, 124–137 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, Y., An, R.: Temporal error analysis of a new Euler semi-implicit scheme for the incompressible Navier-Stokes equations with variable density. Commun. Nonlinear Sci. Numer. Simul. 109, 106330 (2022)

    Article  MathSciNet  Google Scholar 

  23. Li, Y., Li, C.Y., Cui, X.W.: Spatial error analysis of a new Euler finite element scheme for the incompressible flows with variable density, submitted, (2022)

  24. Lions, P.L.: Mathematical Topics in Fluid Mechanics, Volume 1: Incompressible Models, Oxford University Press, Oxford, UK, (1996)

  25. Liu, C., Walkington, N.J.: Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity. SIAM J. Numer. Anal. 45, 1287–1304 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, J.: Simple and efficient ALE methods with provable temporal accuracy up to fifth order for the Stokes equations on time varying domains. SIAM J. Numer. Anal. 51, 743–772 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications, vol. 69. Springer, Berlin Heidelberg (2012)

    MATH  Google Scholar 

  28. Pyo, J.H., Shen, J.: Gauge-Uzawa methods for incompressible flows with variable density. J. Comput. Phys. 221, 181–197 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Temam, R.: Sur l’approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires. II. Arch. Rational Mech. Anal. 33, 377–385 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  30. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, New York (2006)

    MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 11771337) and Natural Science Foundation of Zhejiang Province (No. LY23A010002).

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  1. College of Mathematics and Physics, Wenzhou University, Wenzhou, People’s Republic of China

    Yuan Li & Rong An

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  1. Yuan Li
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  2. Rong An
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Correspondence to Rong An.

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Li, Y., An, R. Error Analysis of a Unconditionally Stable BDF2 Finite Element Scheme for the Incompressible Flows with Variable Density. J Sci Comput 95, 73 (2023). https://doi.org/10.1007/s10915-023-02205-6

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