Duan, Mengmeng; Yang, Yan; Feng, Minfu A weak Galerkin finite element method for the Kelvin-Voigt viscoelastic fluid flow model. (English) Zbl 1505.65261 Appl. Numer. Math. 184, 406-430 (2023). Summary: In this paper, we consider a weak Galerkin finite element method for the Kelvin-Voigt viscoelastic fluid flow model. Firstly, the weak Galerkin finite element method is used to approximate the spatial variable and we use piecewise polynomials of degrees \(k\), \(k - 1\) and \(k - 1\) (\(k \geq 1\)) to approximate the velocity, pressure, and the numerical trace of the velocity on the interfaces of elements, respectively. Secondly, the backward Euler difference method is adopted in the temporal discretization for the fully discrete scheme. Furthermore, the stability and optimal convergence of numerical solutions in \(L^\infty( L^2)\) and \(L^\infty( H^1)\)-norms of velocity as well as \(L^\infty( L^2)\)-norm of pressure were presented. Finally, numerical examples verify the effectiveness of the proposed method, which also obtain that the algorithm has convergence and robust for different retardation time. Cited in 3 Documents MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 76A10 Viscoelastic fluids 76M10 Finite element methods applied to problems in fluid mechanics 76M20 Finite difference methods applied to problems in fluid mechanics 35Q35 PDEs in connection with fluid mechanics Keywords:weak Galerkin; finite element method; Kelvin-Voigt model; error estimates × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Bajpai, S.; Nataraj, N.; Pani, A. K.; Damazio, P.; Yuan, J. Y., Semidiscrete Galerkin method for equations of motion arising in Kelvin-Voigt model of viscoelastic fluid flow, Numer. Methods Partial Differ. Equ., 29, 3, 857-883 (2013) · Zbl 1266.76028 [2] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag · Zbl 0788.73002 [3] Burtscher, M.; Szczyrba, I., Numerical modeling of brain dynamics in traumatic situations-impulsive translations, (The 2005 International Conference on Mathematics and Engineering Techniques in Medicine and Biological Sciences (2005)), 205-211 [4] Burtscher, M.; Szczyrba, I., Computational simulation and visualization of traumatic brain injuries, (2006 International Conference on Modeling. Simulation and Visualization Methods (2006)), 101-107 [5] Cao, Y. P.; Lunasin, E. M.; Titi, E. S., Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4, 4, 823-848 (2006) · Zbl 1127.35034 [6] Chen, G.; Feng, M. F.; Xie, X. P., Robust globally divergence-free weak Galerkin methods for Stokes equations, J. Comput. Math., 34, 5, 549-572 (2016) · Zbl 1389.76027 [7] Cockburn, B.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comput., 52, 186, 411-435 (1989) · Zbl 0662.65083 [8] Cotter, C. S.; Smolarkiewicz, P. K.; Szczyrba, I. N., A viscoelastic fluid model for brain injuries, Int. J. Numer. Methods Fluids, 40, 303-311 (2002) · Zbl 1058.76532 [9] Di Pietro, D. A.; Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods (2012), Springer-Verlag · Zbl 1231.65209 [10] Gao, F. Z.; Wang, X. S., A modified weak Galerkin finite element method for a class of parabolic problems, J. Comput. Appl. Math., 271, 1, 1-19 (2014) · Zbl 1321.65154 [11] Girault, V.; Raviart, P. A., Finite Element Methods for Navier-Stokes Equations (1986), Springer Berlin Heidelberg · Zbl 0585.65077 [12] Heywood, J. G.; Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19, 2, 275-311 (1982) · Zbl 0487.76035 [13] Heywood, J. G.; 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, 2, 353-384 (1990) · Zbl 0694.76014 [14] Hill, A. T.; Süli, E., Approximation of the global attractor for the incompressible Navier-Stokes equations, IMA J. Numer. Anal., 20, 4, 633-667 (2000) · Zbl 0982.76022 [15] Hu, X. Z.; Mu, L.; Ye, X., A weak Galerkin finite element method for the Navier-Stokes equations, J. Comput. Appl. Math., 362, 614-625 (2019) · Zbl 1422.65389 [16] 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 [17] 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) [18] Lin, G.; Liu, J.; Sadre-Marandi, F., A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods, J. Comput. Appl. Math., 273, 346-362 (2015) · Zbl 1295.65114 [19] Liu, X.; Li, J.; Chen, Z. X., A weak Galerkin finite element method for the Navier-Stokes equations, J. Comput. Appl. Math., 333, 442-457 (2018) · Zbl 1395.76040 [20] Mu, L.; Wang, J. P.; Wang, Y. Q.; Ye, X., A computational study of the weak Galerkin method for second-order elliptic equations, Numer. Algorithms, 63, 4, 753-777 (2013) · Zbl 1271.65140 [21] Mu, L.; Wang, J. P.; Ye, X., Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12, 1, 31-53 (2015) · Zbl 1332.65172 [22] Mu, L.; Wang, J. P.; Ye, X.; Zhang, S. Y., A discrete divergence free weak Galerkin finite element method for the Stokes equations, Appl. Numer. Math., 125, 172-182 (2018) · Zbl 1378.76051 [23] Oskolkov, A. P., Uniqueness and global solvability for boundary-value problems for the equations of motion of water solutions of polymers, Zap. Nauč. Semin. POMI, 38, 98-136 (1973) [24] Oskolkov, A. P., Theory of nonstationary flows of Kelvin-Voigt fluids, J. Sov. Math., 28, 5, 751-758 (1985) · Zbl 0561.76017 [25] Oskolkov, A. P., Initial-boundary value problems for equations of motion of Kelvin-Voight fluids and Oldroyd fluids, Tr. Mat. Inst. Steklova, 179, 126-164 (1988) · Zbl 0674.76004 [26] Oskolkov, A. P.; Shadiev, R. D., Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids, J. Sov. Math., 62, 2, 2699-2723 (1992) · Zbl 0784.76006 [27] Oskolkov, A. P.; Shadiev, R. D., 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. Math. Sci., 68, 2, 240-253 (1994) · Zbl 0850.76039 [28] Pany, A. K., Fully discrete second-order backward difference method for Kelvin-Voigt fluid flow model, Numer. Algorithms, 78, 4, 1061-1086 (2018) · Zbl 1408.76369 [29] Pany, A. K.; Bajpai, S.; Pani, A. K., 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 (2016) · Zbl 1381.76194 [30] Pavlovskii, V. A., Theoretical description of weak aqueous polymer solutions, Sov. Phys. Dokl., 16, 853 (1971) [31] Teschl, G., Ordinary Differential Equations and Dynamical Systems (2012), American Mathematical Soc. · Zbl 1263.34002 [32] Wang, J. P.; Ye, X., A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241, 1, 103-115 (2013) · Zbl 1261.65121 [33] Wang, J. P.; Ye, X., A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comput., 83, 289, 2101-2126 (2014) · Zbl 1308.65202 [34] Wang, J. P.; Ye, X., A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42, 1, 155-174 (2016) · Zbl 1382.76178 [35] Zhang, B. J.; Yang, Y.; Feng, M. F., A \(C^0\)-weak Galerkin finite element method for the two-dimensional Navier-Stokes equations in stream-function formulation, J. Comput. Math., 38, 2, 310 (2020) · Zbl 1463.65383 [36] Zhang, T.; Duan, M. M., One-level and multilevel space-time finite element method for the viscoelastic Kelvin-Voigt model, Math. Methods Appl. Sci., 43, 7, 4744-4768 (2020) · Zbl 1446.65122 [37] Zhang, T.; Duan, M. M., Stability and convergence analysis of stabilized finite element method for the Kelvin-Voigt viscoelastic fluid flow model, Numer. Algorithms, 87, 3, 1201-1228 (2021) · Zbl 1476.65262 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.