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A large time stepping viscosity-splitting finite element method for the viscoelastic flow problem

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Abstract

In this article, a large time stepping viscosity-splitting scheme is considered for the viscoelastic flow problem. The temporal term is decomposed into a sequence of two steps (using decomposition of the viscosity). For the first step, a linear elliptic problem is solved with explicit scheme for the convection term (a linear system with a constant coefficient matrix is obtained and the computation becomes easy). At the second step, the problem has the structure of the Stokes problem. Both two problems satisfy the homogeneous Derichlet boundary conditions for the velocities. The main novelties of this work are the stability of numerical solutions under the condition k 1 Δt ≤ 1 with a positive constant k 1, and optimal error estimates for both velocity in L ∞(H 1) and L 2(H 1) norms and pressure in L ∞(L 2) and L 2(L 2) norms. In order to enlarge the time step, we introduce a diffusion term θ Δu in all steps of the schemes. Finally, some numerical results are provided to display the performance of our algorithm.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Blasco, J., Codina, R., Huerta, A.: A fractional-step method for the incompressible Navier-Stokes equations related to a predictor-multicorrector algorithm. Int. J. Numer. Methods Fluids 28, 1391–1419 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Chorin, A.J.: On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comput. 23, 341–353 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  5. Codina, R.: Pressure stability in fractional step finite element methods for incompressible flows. J. Comput. Phys. 170, 112–140 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Codina, R., Badia, S.: On some pressure segregation methods of fractional-step type for the finite element approximation of incompressible flow problems. Comput. Methods Appl. Mech. Eng. 195, 2900–2918 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dukowicz, J.K., Dvinsky, A.S.: Approximate factorization as a high order splitting for the implicit incompressible flow equations. J. Comput. Phys. 102, 336–347 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Feng, X.L., He, Y.N., Liu, D.M.: Convergence analysis of an implicit fractional-step method for the incompressible Navier-Stokes equations. Appl. Math. Model. 35, 5856–5871 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fernandez-Cara, E., Marin Beltran, M.: The convergence of two numerical schemes for the Navier-Stokes equations. Numer. Math. 55, 33–60 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gervasio, P., Saleri, F.: Algebraic fractional-step schemes for time-dependent incompressible Navier-Stokes equations. J. Sci. Comput. 27, 257–269 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gervasio, P., Saleri, F., Veneziani, A.: Algebraic fractional-step schemes with spectral methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 214, 347–365 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Girault, V., Raviart, P.A.: Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin (1987)

    Google Scholar 

  13. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems, 2nd edn.Springer-Verlag, New York (1984)

    Book  MATH  Google Scholar 

  14. Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engng. 195, 6011–6045 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Guermond, J.L., Quartapelle, L.: On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80, 207–238 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Guillen-Gonzalez, F., Redondo-Neble, M.V.: New error estimates for a viscosity splitting scheme in time for the three-dimensional Navier-Stokes equations. IMA J. Numer. Anal. 31, 556–579 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59, 308–323 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  18. He, Y.N.: The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Math. Comput. 77, 2097–2124 (2008)

    Article  MATH  Google Scholar 

  19. He, Y.N.: Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. He, Y.N., Li, J.: A penalty fintie element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations. J. Comput. Appl. Math. 235, 708–725 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. He, Y.N., Lin, Y.P., Shen, S.S.P., Sun, W.W., Tait, R.: Finite element approximation for the viscoelastic fluid motion problem. J. Comput. Appl. Math. 155, 201–222 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. He, Y.N., Liu, Y.X., Tang, T.: On large time-stepping methods for the Cahn-Hilliard equation. Appl. Numer. Math. 57, 616–628 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Heywood, J., Rannacher, R.: Finite element approximation of the non-stationary Navier-Stokes problem I: Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal 19, 275–311 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  24. Oldroyd, J.: Some steady flows of the general elastic-viscous liquid. Proc. Roy. Soc. Ser. A. 283, 115–133 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pani, A., Yuan, J.Y., Damazio, P.: On a linear linearized backward Euler method for the equations of motion arising in the Oldroyd model. IMA J. Numer. Anal. 25, 750–782 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Quarteroni, A., Saleri, F., Veneziani, A.: Factorization methods for the numerical approximation of Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 188, 505–526 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Saleri, F., Veneziani, A.: Pressure correction algebraic splitting methods for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 43, 174–194 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Shen, J.: On error estimates of projection methods for Navier-Stokes equations: First order schemes. SIAM J. Numer. Anal. 29, 57–77 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  29. Shen, J.: Remarks on the pressure error estimates for the projection methods. Numer. Math. 67, 513–520 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  30. Strikwerda, J.C., Young, S.L.: The accuracy of the fractional step method. SIAM J. Numer. Anal. 37, 37–47 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sobolevskii, P.E.: Stabilization of viscoelastic fluid motion, Oldroyd’s mathematical model. Diff. Integr. Equ. 7, 1579–1621 (1994)

    MathSciNet  Google Scholar 

  32. Temam, R.: Navier-Stokes equations. In: Theory and Numerical Analysis, 3rd edn. North-Holland, Amsterdam (1984)

    Google Scholar 

  33. Temam, R.: Remark on the pressure boundary condition for the projection method. Theor. Comput. Fluid Dyn. 3, 181–184 (1991)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, 454003, Jiaozuo, People’s Republic of China

    Tong Zhang

  2. Departamento de Matemática, Universidade Federal do Paraná, Centro Politécnico, Curitiba, 81531-980, People’s Republic of Brazil

    Damazio Pedro & JinYun Yuan

Authors
  1. Tong Zhang
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  2. Damazio Pedro
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  3. JinYun Yuan
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Correspondence to Tong Zhang.

Additional information

Communicated by: John Lowengrub

This work was partially supported by CAPES and CNPq, Brazil, and the work of the first author was supported by NSF of China (No. 11301157), the Natural Science Foundation of the Education Department of Henan Province (No.14A110008) and the Doctor Fund of Henan Polytechnic University (B2012-098).

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Zhang, T., Pedro, D. & Yuan, J. A large time stepping viscosity-splitting finite element method for the viscoelastic flow problem. Adv Comput Math 41, 149–190 (2015). https://doi.org/10.1007/s10444-014-9353-4

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  • DOI: https://doi.org/10.1007/s10444-014-9353-4

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