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The Euler implicit/explicit FEM for the Kelvin–Voigt model based on the scalar auxiliary variable (SAV) approach

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Abstract

The objective of this paper is to establish the unconditional stability results of Euler implicit/explicit scheme for the viscoelastic Kelvin–Voigt model with a scalar auxiliary variable. We first reformulate the Kelvin–Voigt model into an equivalent system with three variables. The standard Galerkin finite element method is used to approximate the spatial discretization. Then the Euler implicit/explicit method is adopted to discrete the considered problem, a constant coefficient algebraic system is formed and it can be solved efficiently. The unconditional energy dissipation and stability results of numerical solutions in various norms are established. Optimal error estimates are also presented by the energy method and the Gronwall lemma. Finally, some numerical results are given to verify the established theoretical findings and show the performances of the considered numerical scheme.

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References

  • Ammi AA, Marion M (1994) Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier–Stokes equations. Numer Math 68:189–213

    Article  MathSciNet  Google Scholar 

  • Bajpai S, Nataraj N (2014) On a two-grid finite element scheme combined with Crank–Nicolson method for the equations of motion arising in the Kelvin–Voigt model. Comput Math Appl 86:2277–2291

    Article  MathSciNet  Google Scholar 

  • Bajpai S, Nataraj N, Pani AK (2013a) On fully discrete finite element schemes for equations of motion of Kelvin–Voigt fluids. Int J Numer Anal Model 10:481–507

    MathSciNet  MATH  Google Scholar 

  • Bajpai S, Nataraj N, Pani AK, Damazio P, Yuan JY (2013b) Semidiscrete Galerkin method for equations of motion arising in Kelvin–Voigt model of viscoelastic fluid flow. Numer Methods Partial Differ Equ 29:857–883

    Article  MathSciNet  Google Scholar 

  • Bajpai S, Nataraj N, Pani AK (2014) On a two-grid finite element scheme for the equations of motion arising in Kelvin–Voigt model. Adv Comput Math 40:1043–1071

    Article  MathSciNet  Google Scholar 

  • Dubois T, Jauberteau F, Temam R (1993) Solution of the incompressible Navier–Stokes equations by the nonlinear Galerkin method. J Sci Comput 8:167–194

    Article  MathSciNet  Google Scholar 

  • Frutos JD, Garcia-Archilla B, Novo J (2011) Nonlinear convection–diffusion problems: fully discrete approximations and a posteriori error estimates. IMA J Numer Anal 31:1402–1430

    Article  MathSciNet  Google Scholar 

  • Girault V, Raviart PA (1986) Finite element methods for the the Navier–Stokes equations. Spinger, Berlin

    Book  Google Scholar 

  • Hansen E, Stillfjord T (2013) Convergence of the implicit-explicit Euler scheme applied to perturbed dissipative evolution equations. Math Comput 82(284):1975–1985

    Article  MathSciNet  Google Scholar 

  • He YN (2008) The Euler implicit/explicit scheme for the 2D time-dependent Navier–Stokes equations with smooth or non-smooth initial data. Math Comput 77(264):2097–2124

    Article  MathSciNet  Google Scholar 

  • He YN (2015) Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. IMA J Numer Anal 35:767–801

    Article  MathSciNet  Google Scholar 

  • He YN, Li KT (2005) Two-level stabilized finite element methods for the steady Navier–Stokes problem. Computing 74:337–351

    Article  MathSciNet  Google Scholar 

  • He YN, Li J (2010) A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier–Stokes equations. J Comput Appl Math 235(3):708–725

    Article  MathSciNet  Google Scholar 

  • He YN, Huang PZ, Feng XL (2015) \(H^2\)-stability of first order fully discrete schemes for the time-dependent Navier–Stokes equations. J Sci Comput 62:230–264

    Article  MathSciNet  Google Scholar 

  • Hou T, Shi ZQ (2008) An efficient semi-implicit immersed boundary method for the Navier–Stokes equations. J Comput Phys 227:8968–8991

    Article  MathSciNet  Google Scholar 

  • Kaya S, Rivière B (2006) A two-grid stabilization method for solving the steady-state Navier–Stokes equations. Numer Methods Partial Differ Equ 22:728–743

    Article  MathSciNet  Google Scholar 

  • Ladyzenskaya AP (1969) The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York

    Google Scholar 

  • Marti J, Ryzhakov PB (2019) An explicit-implicit finite element model for the numerical solution of incompressible Navier–Stokes equations on moving grids. Comput Methods Appl Mech Eng 350:750–765

    Article  MathSciNet  Google Scholar 

  • Oskolkov AP (1985) Theory of nonstationary flows of Kelvin–Voigt fluids. J Sov Math 28:751–758

    Article  Google Scholar 

  • Oskolkov AP (1992) On an estimate, uniform on the semiaxis \(t\ge 0\), for the rate of convergence of Galerkin approximations for the equations of motion of Kelvin–Voight fluids. J Sov Math 62:2802–2806

    Article  MathSciNet  Google Scholar 

  • Oskolkov AP, Shadiev RD (1994) Towards a theory of global solvability on \([0,\infty ]\) of initial-boundary value problems for the equations of motion of Oldroyd and Kelvin–Voight fluids. J Sov Math 68:240–253

    MATH  Google Scholar 

  • Pani AK, Pany AK, Damazio P, Yuan JY (2014) A modified nonlinear spectral Galerkin method for the equations of motion arising in the Kelvin–Voigt fluids. Appl Anal 93:1587–1610

    Article  MathSciNet  Google Scholar 

  • Pany AK, Bajpai S, Pani AK (2016) Optimal error estimates for semidiscrete Galerkin approximations to equations of motion described by Kelvin–Voigt viscoelastic fluid flow model. J Comput Appl Math 302:234–257

    Article  MathSciNet  Google Scholar 

  • Rautmann R (1980) On the convergence-rate of nonstationary Navier–Stokes approximations. Approx Methods Navier Stokes Probl 771:425–449

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Shen J (1992b) On error estimates of some higher order projection and penalty-projection methods for Navier–Stokes equations. Numer Math 62:49–73

    Article  MathSciNet  Google Scholar 

  • Shen J (1996) On error estimates of the projection, methods for the Navier–Stokes equations: second-order schemes. Math Comput 65:1039–1066

    Article  MathSciNet  Google Scholar 

  • Shen J, Xu J (2018) Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J Numer Anal 56:2895–2912

    Article  MathSciNet  Google Scholar 

  • Shen J, Xu J, Yang J (2018) The scalar auxiliary variable (SAV) approach for gradient flows. J Comput Phys 353:407–416

    Article  MathSciNet  Google Scholar 

  • Temam R (1984) Navier–Stokes equations, theory and numerical analysis, 3rd edn. North-Holland, Amsterdam

    MATH  Google Scholar 

  • Zhang T, Duan MM (2020) One-level and multilevel space-time finite element method for the viscoelastic Kelvin–Voigt model. Math Methods Appl Sci 43:4744–4768

    MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

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

    Jinting Yang

  2. School of Mathematics and Information Science, Yantai University, Yantai, 264000, People’s Republic of China

    Tong Zhang

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  1. Jinting Yang
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  2. Tong Zhang
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Correspondence to Tong Zhang.

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Communicated by Pablo Pedregal.

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This work is supported by the NSF of China (no. 11971152) and NSF of Henan Province (202300410167)

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Yang, J., Zhang, T. The Euler implicit/explicit FEM for the Kelvin–Voigt model based on the scalar auxiliary variable (SAV) approach. Comp. Appl. Math. 40, 133 (2021). https://doi.org/10.1007/s40314-021-01517-3

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  • DOI: https://doi.org/10.1007/s40314-021-01517-3

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