Previous Articles     Next Articles

MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS

Xiaocui Li1, Xu You2   

  1. 1. College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China;
    2. Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing 102617, China
  • Received:2018-07-18 Revised:2018-07-30 Online:2021-01-15 Published:2021-03-11

Xiaocui Li, Xu You. MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS[J]. Journal of Computational Mathematics, 2021, 39(1): 130-146.

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations. The proposed method is based on the mixed finite element method in space and a finite difference scheme in time. The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail. Furthermore, We give the convergence analysis for both semidiscrete and fully discrete schemes and then prove that the numerical solution converges the exact one with order O(h2 + k), where h and k respectively denote the space step size and the time step size. Finally, numerical examples are presented to demonstrate the effectiveness of our numerical methods.

CLC Number: 

[1] G.A. Baker, Galerkin approximations for the Navier-Stokes equations, Manuscript, Harvard University, Cambridge, MA, 1976.
[2] C. Bernardi and G. Raugel, A conforming finite element method for the time-dependent Navier-Stokes equations, Siam J. Numer. Anal., 22:3(1985), 455-473.
[3] D.L. Brown, R. Cortez and M. L. Minion, Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168:2(2001), 464-499.
[4] E. Burman, Pressure projection stabilizations for Galerkin approximations of Stokes and Darcys problem, Numer. Meth. Part. D. E., 24(2008), 127-143.
[5] W. Bu, Y. Tang and J. Yang, Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276(2014), 26-38.
[6] W.H. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47:1(2008), 204-226.
[7] W.H. Deng and J.S. Hesthaven, Local discontinuous Galerkin methods for fractional ordinary differential equations, BIT, 55(2015), 967-985.
[8] C. Févrière, J. Laminie, P. Poullet and P. Poullet, On the penalty-projection method for the Navier-Stokes equations with the MAC mesh, J. Comput. Appl. Math., 226(2009), 228-245.
[9] J.D. Frutos, B. Garca-Archilla and J. Novo, Optimal error bounds for two-grid schemes applied to the Navier-Stokes equations, Appl. Math. Comput., 218:13(2012), 7034-7051.
[10] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms, Springer-Verlag, Berlin Heidelberg, 1986.
[11] D. Goswami and P.D. Damázio, A two-level finite element method for time-dependent incompressible Navier-Stokes equations with non-smooth initial data, arXiv:1211.3342[math.NA].
[12] B.Y. Guo and Y.J. Jiao, Spectral method for Navier-Stokes equations with slip boundary conditions, J. Sci. Comput. 58(2014), 249-274.
[13] J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary NavierStokes problem. I. Regularity of solutions and second-order spatial discretization, SIAM J. Numer. Anal., 19(1982), 275-311.
[14] Y.N. He and W.W. Sun, Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., 45:2(2007), 837-869.
[15] Y.N. He and J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations, Comput. Method. Appl. M., 198(2009), 1351-1359.
[16] Y.N. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., 41(2004), 1263-1285.
[17] P. Huang, X. Feng and D. Liu, A stabilized finite element method for the time-dependent Stokes equations based on Crank-Nicolson Scheme, Appl. Math. Model., 37:4(2013), 1910-1919.
[18] Y. He, P. Huang and X. Feng, H2-stability of the first order fully discrete schemes for the timedependent Navier-Stokes equations, J. Sci. Comput., 62:1(2015), 230-264.
[19] H. Johnston and J.G. Liu, Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, J. Comput. Phys., 199:1(2004), 221-259.
[20] Y. Jiang and J. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235:11(2011), 3285-3290.
[21] B. Jin, R. Lazarov and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51:1(2012), 445-466.
[22] B. Jin, R. Lazarov, J. Pasciak and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA J. Numer. Anal., 2014.
[23] B. Jin, R. Lazarov and Y. Liu, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281(2015), 825-843.
[24] B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 2015.
[25] J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59:2(1985), 308-323.
[26] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[27] Li X, Yang X, Zhang Y., Error Estimates of mixed finite element methods for time-fractional Navier-Stokes equations, J. Sci. Comput., 70:2(2017), 500-515.
[28] S. Karaa, K. Mustapha and A.K. Pani, Finite volume element method for two-dimensional fractional subdiffusion problems, IMA J. Numer. Anal., (2015), 1-17.
[29] Y. Lin and L. Wahlbin, Ritz-Volterra projections to finite-element spaces and applications to integrodifferential and related equations, SIAM J. Numer. Anal., 28:4(1991), 1047-1070.
[30] F. Liu, V. Anh, I. Turner and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput., 13(2003), 233-245.
[31] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225:2(2007), 1533-1552.
[32] Q. Liu and Y. Hou, A two-level finite element method for the Navier-Stokes equations based on a new projection, Appl. Math. Model., 34:2(2010), 383-399.
[33] C.P. Li, Z.G. Zhao and Y.Q. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62.3(2011), 855-875.
[34] Z.D. Luo, A new finite volume element formulation for the non-stationary Navier-Stokes equations, Adv. Appl. Math. Mech., 6(2014), 615-636.
[35] Y. Liu, Y. Du, H. Li and J. Li, A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative, Comput. Math. Appl., 70:10(2015), 2474-2492.
[36] Y. Liu, Y. Du, H. Li, S. He and W. Gao, Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem, Comput. Math. Appl., 70:4(2015), 573-591.
[37] C.H. Min and F. Gibou, A second order accurate projection method for the incompressible NavierStokes equations on non-graded adaptive grids, J. Comput. Phys., 219(2006), 912-929.
[38] R.H. Nochetto and J.H. Pyo, The Gauge-Uzawa finite element method. Part I:The Navier-Stokes equations, SIAM J. Numer. Anal., 43(2005), 1043-1068.
[39] H. Okamoto, On the semi-discrete finite element approximation for the nonstationary NavierStokes equation, Journal of the Faculty of Science the University of Tokyo. sect A Math., 29:3(1982), 613-651.
[40] A.K. Pani and J.Y. Yuan, Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model, IMA J. Numer. Anal., 25:4(2005), 750-782.
[41] R. Rannacher, Numerical analysis of the Navier-Stokes equations, Appl. Math., 38(1993), 361-380.
[42] T. Chac′on Rebollo, M. G′omez Mármol and M. Restelli, Numerical analysis of penalty stabilized finite element discretizations of evolution Navier-Stokes equations, J. Sci. Comput., 63(2015), 885-912.
[43] J. Shen, On error estimates of projection methods for the Navier-Stokes equations:second order schemes, Math. Comput., 65(1996), 1039-1065.
[44] L. Shan and Y. Hou, A fully discrete stabilized finite element method for the time-dependent Navier-Stokes equations, Appl. Math. Comput., 215:1(2009), 85-99.
[45] S.S. Siddiqi and S. Arshed, Numerical solution of time-fractional fourth-order partial differential equations, Int. J. Comput. Math., 5(2014), 1-23.
[46] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.
[47] F. Tone, Error analysis for a second order scheme for the Navier-Stokes equations, Appl. Numer. Math., 50:1(2004), 93-119.
[48] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Spriger Series in Computational Mathematics vol. 25, Springer-Verlag, Berlin Heidelberg, 1997.
[49] MF. Wheeler, A Priori L2 Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations, SIAM J. Numer. Anal., 10:10(1973), 723-759.
[50] P. Zhuang, F. Liu, V. Anh and I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., 46(2008), 1079-1095.
[51] F.H. Zeng, C.P. Li, F.W. Liu and I. Turner, Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM J. Sci. Comput., 37(2015), 55-78.
[52] X.d. Zhang, J. Liu, L. Wei and C. Ma, Finite element method for GrwnwaldCLetnikov timefractional partial differential equation, Appl. Anal., 92(2013), 2103-2114.
[53] P. Zhuang, F. Liu, I. Turner and V. Anh, Galerkin finite element method and error analysis for the fractional cable equation, Numer. Algorithms, (2015), 1-20.
[54] Lijing Zhao, Weihua Deng and Jan S Hesthaven, Spectral Methods for Tempered Fractional Differential Equations, 2016.
[55] G.A. Zou, G.Y. Lv and J.L. Wu, Stochastic Navier-Stokes equations with Caputo derivative driven by fractional noises, J. Math. Anal. Appl., 461:1(2017), 595-609.
[56] G.A. Zou, Y.Zhou, Ahmad B, et al., Finite difference/element method for time-fractional NavierStokes equations, 2018.
[57] M. Gunzburger, B. Li and J. Wang, Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise, Math. Comput., (2017), DOI:10.1090/mcom/3397.
[1] Kaibo Hu, Ragnar Winther. WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS [J]. Journal of Computational Mathematics, 2021, 39(3): 333-357.
[2] Xiaodi Zhang, Weiying Zheng. MONOLITHIC MULTIGRID FOR REDUCED MAGNETOHYDRODYNAMIC EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(3): 453-470.
[3] Xiaoliang Song, Bo Chen, Bo Yu. ERROR ESTIMATES FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION [J]. Journal of Computational Mathematics, 2021, 39(3): 471-492.
[4] Michael Holst, Yuwen Li, Adam Mihalik, Ryan Szypowski. CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK [J]. Journal of Computational Mathematics, 2020, 38(5): 748-767.
[5] Qilong Zhai, Xiaozhe Hu, Ran Zhang. THE SHIFTED-INVERSE POWER WEAK GALERKIN METHOD FOR EIGENVALUE PROBLEMS [J]. Journal of Computational Mathematics, 2020, 38(4): 606-623.
[6] Weifeng Zhang, Shuo Zhang. ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS [J]. Journal of Computational Mathematics, 2020, 38(4): 565-579.
[7] Juncai He, Lin Li, Jinchao Xu, Chunyue Zheng. RELU DEEP NEURAL NETWORKS AND LINEAR FINITE ELEMENTS [J]. Journal of Computational Mathematics, 2020, 38(3): 502-527.
[8] Jie Chen, Zhengkang He, Shuyu Sun, Shimin Guo, Zhangxin Chen. EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS [J]. Journal of Computational Mathematics, 2020, 38(3): 452-468.
[9] Li Cai, Ye Sun, Feifei Jing, Yiqiang Li, Xiaoqin Shen, Yufeng Nie. A FULLY DISCRETE IMPLICIT-EXPLICIT FINITE ELEMENT METHOD FOR SOLVING THE FITZHUGH-NAGUMO MODEL [J]. Journal of Computational Mathematics, 2020, 38(3): 469-486.
[10] Huoyuan Duan, Roger C. E. Tan. ERROR ANALYSIS OF A STABILIZED FINITE ELEMENT METHOD FOR THE GENERALIZED STOKES PROBLEM [J]. Journal of Computational Mathematics, 2020, 38(2): 254-290.
[11] Nikolaus von Daniels, Michael Hinze. VARIATIONAL DISCRETIZATION OF A CONTROL-CONSTRAINED PARABOLIC BANG-BANG OPTIMAL CONTROL PROBLEM [J]. Journal of Computational Mathematics, 2020, 38(1): 14-40.
[12] Carsten Carstensen, Sophie Puttkammer. HOW TO PROVE THE DISCRETE RELIABILITY FOR NONCONFORMING FINITE ELEMENT METHODS [J]. Journal of Computational Mathematics, 2020, 38(1): 142-175.
[13] Yu Du, Haijun Wu, Zhimin Zhang. SUPERCONVERGENCE ANALYSIS OF THE POLYNOMIAL PRESERVING RECOVERY FOR ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS [J]. Journal of Computational Mathematics, 2020, 38(1): 223-238.
[14] Weijie Huang, Zhiping Li. A MIXED FINITE ELEMENT METHOD FOR MULTI-CAVITY COMPUTATION IN INCOMPRESSIBLE NONLINEAR ELASTICITY [J]. Journal of Computational Mathematics, 2019, 37(5): 609-628.
[15] Li Guo, Hengguang Li, Yang Yang. INTERIOR ESTIMATES OF SEMIDISCRETE FINITE ELEMENT METHODS FOR PARABOLIC PROBLEMS WITH DISTRIBUTIONAL DATA [J]. Journal of Computational Mathematics, 2019, 37(4): 458-474.
Viewed
Full text


Abstract