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A modified characteristic projection finite element method for the Kelvin-Voigt viscoelastic fluid equations. (English) Zbl 1524.76227

Summary: This paper presents a modified characteristic projection finite element method for the Kelvin-Voigt viscoelastic fluid equations. Then, the unconditional stability and optimal convergence of numerical solutions in \(L^2\)-norms and \(H^1\)-norms were presented. In addition, some numerical results will be given to testify the theoretical analysis. The numerical results show that the convergence orders are optimal, which show the theoretical analysis is right. The numerical results show that our method is robust for different Reynolds numbers. The numerical results imply that the numerical algorithm also converges for different retardation time. It means that our numerical method is robust for different retardation time.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76A10 Viscoelastic fluids

Software:

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References:

[1] Oskolkov, A. P., The uniqueness and global solvability for boundary value problems for the questions of motion of water solutions of polymers, Zap. Nauč. Semin. POMI, 38, 98-136 (1973)
[2] Burtscher, M.; Szczyrba, I., Numerical modeling of brain dynamics in traumatic situation-impulsive translations, (The 2005 International Conference on Mathematics and Engineering Techniques in Medicine and Biological Sciences (2005)), 205-211
[3] Burtscher, M.; Szczyrba, I., Computational simulation and visualization of traumatic brain injuries, (2006 International Conference of Modeling, Simulation and Visualization Methods (2006)), 101-107
[4] Hong, G.; Hong, H., Logarithmic stabilization of the Kirchhoff plate transmission system with locally distributed Kelvin-Voigt damping, Appl. Math., 1-27 (2021)
[5] Kundu, S.; Bajpai, S.; Pani, A. K., Asymptotic behavior and finite element error estimates of Kelvin-Voigt viscoelastic fluid flow model, Numer. Algorithms, 75, 3, 1-35 (2017) · Zbl 1393.76066
[6] Pani, A. K.; Pany, A. K.; Damazio, P.; Yuan, J., A modified nonlinear spectral Galerkin method for the equations of motion arising in the Kelvin-Voigt fluids, Appl. Anal., 93, 8, 1587-1610 (2014) · Zbl 1356.76218
[7] Zhang, T.; Duan, M., One-level and multilevel space-time finite element method for the viscoelastic Kelvin-Voigt model, Math. Methods Appl. Sci., 43, 7, 1-25 (2020)
[8] Zhang, T.; Duan, M., Stability and convergence analysis of stabilized finite element method for the Kelvin-Voigt viscoelastic fluid flow model, Numer. Algorithms, 87, 1201-1228 (2021) · Zbl 1476.65262
[9] Kadchenko, S. I.; Kondyukov, A. O., Numerical study of a flow of viscoelastic fluid of Kelvin-Voigt having zero order in a magnetic field, J. Comput. Eng. Math., 3, 2, 40-47 (2016) · Zbl 1455.76208
[10] Yang, J.; Zhang, T., The Euler implicit/explicit FEM for the Kelvin-Voigt model based on the scalar auxiliary variable (SAV) approach, Comput. Appl. Math., 40, 4, 1-18 (2021) · Zbl 1476.65258
[11] Kondyukov, A. O.; Sukacheva, T. G., Non-stationary model of incompressible viscoelastic Kelvin-Voigt fluid of higher order in the Earth’s magnetic field, J. Phys. Conf. Ser., 1658, 1, Article 012028 pp. (2020)
[12] Temam, R., Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II), Arch. Ration. Mech. Anal., 32, 2, 135-153 (1969) · Zbl 0195.46001
[13] Chorin, J., Numerical solution of the Navier-Stokes equations, Comput. Fluid Mech., 22, 104, 745-762 (1968) · Zbl 0198.50103
[14] Chorin, J., On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comput., 23, 106, 341 (1969) · Zbl 0184.20103
[15] Guermond, J.; Shen, J., On the error estimates for the rotational pressure-correction projection methods, Math. Comput., 73, 248, 1719-1738 (2003) · Zbl 1093.76050
[16] Shen, J., On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes, Math. Comput., 65, 1039-1065 (1996) · Zbl 0855.76049
[17] Si, Z.; Wang, J.; Sun, W., Unconditional stability and error estimates of modified characteristics FEMs for the Navier-Stokes equations, Numer. Math., 134, 139-161 (2016) · Zbl 1346.76073
[18] Yang, Y.; Lei, Y.; Si, Z., Unconditional stability and error estimates of the modified characteristics FEMs for the time-dependent viscoelastic Oldroyd flows, Adv. Appl. Math. Mech., 13, 2, 311-332 (2021) · Zbl 1488.76089
[19] Yang, Y.; Si, Z., Unconditional stability and error estimates of the modified characteristics FEMs for the time-dependent incompressible MHD equations, Comput. Math. Appl., 77, 263-283 (2019) · Zbl 1442.65284
[20] Shen, X.; Wang, Y.; Si, Z., A rotational pressure-correction projection methods for unsteady incompressible magnetohydrodynamics equations, Appl. Math. Comput., 387, Article 124488 pp. (2020) · Zbl 1465.76057
[21] Si, Z.; Lei, Y.; Zhang, T., Unconditional optimal error estimate of the projection/Lagrange-Galerkin finite element method for the Boussinesq equations, Numer. Algorithms, 83, 669-700 (2020) · Zbl 1440.65148
[22] Si, Z.; Wang, Y., Modified characteristics projection finite element method for time-dependent conduction-convection problems, Bound. Value Probl., 1, 151-174 (2015) · Zbl 1338.76067
[23] He, Y., Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations, IMA J. Numer. Anal., 35, 767-801 (2015) · Zbl 1312.76061
[24] Galdi, P., An introduction to the mathematical theory of the Navier-Stokes equations, (Steady-State Problems (2011), Springer: Springer New York) · Zbl 1245.35002
[25] Girauit, V.; Raviart, P., Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms (1987), Spring-Verlag: Spring-Verlag Berlin
[26] Hecht, F., New development in FreeFem++, J. Numer. Math., 20, 251-265 (2012) · Zbl 1266.68090
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