American Institute of Mathematical Sciences

June  2015, 20(4): 1189-1212. doi: 10.3934/dcdsb.2015.20.1189

A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations

 1 School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

Received  October 2012 Revised  July 2014 Published  February 2015

In this article, we employ proper orthogonal decomposition (POD) technique to establish a POD-based reduced-order stabilized mixed finite volume element (SMFVE) extrapolation algorithm based on two local Gaussian integrals, parameter-free, and Crank-Nicolson (CN) method with fewer degrees of freedom for the non-stationary Navier-Stokes equations. The error estimates between the POD-based reduced-order SMFVE solutions and the classical SMFVE solutions and the implementation for the POD-based reduced-order SMFVE extrapolation algorithm are provided. A numerical example is used to illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the POD-based reduced-order SMFVE extrapolation algorithm is feasible and efficient for finding numerical solutions for the non-stationary Navier-Stokes equations.
Citation: Zhendong Luo. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1189-1212. doi: 10.3934/dcdsb.2015.20.1189
References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] D. Ahlman, F. Södelundon, J. Jackson, A. Kurdila and W. Shyy, Proper orthogonal decomposition for time-dependent lid-driven cavity flows, Numer. Heat Trans. Part B-Fund., 42 (2002), 285-306. doi: 10.1080/10407790190053950. [3] I. Ammara and C. Masson, Development of a fully coupled control-volume finite element method for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 44 (2004), 621-644. doi: 10.1002/fld.662. [4] J. An, P. Sun, Z. D. Luo and X. M. Huang, A stabilized fully discrete finite volume element formulation for non-stationary Stokes equations, Math. Numer. Sin., 33 (2011), 213-224. [5] R. E. Bank and D. J. Rose, Some error estimates for the box methods, SIAM Journal on Numerical Analysis, 24 (1987), 777-787. doi: 10.1137/0724050. [6] M. Bergmann, C. H. Bruneau and A. Iollo, Enablers for robust POD models, Journal of Computational Physics, 228 (2009), 516-538. doi: 10.1016/j.jcp.2008.09.024. [7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1. [8] Z. Cai and S. McCormick, On the accuracy of the finite volume element method for diffusion equations on composite grid, SIAM Journal on Numerical Analysis, 27 (1990), 636-655. doi: 10.1137/0727039. [9] J. R. Cannon and Y. Lin, A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM Journal on Numerical Analysis, 27 (1990), 595-607. doi: 10.1137/0727036. [10] S. H. Chou and D. Y. Kwak, A covolume method based on rotated bilinears for the generalized Stokes problem, SIAM Journal on Numerical Analysis, 35 (1998), 494-507. doi: 10.1137/S0036142996299964. [11] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [12] A. E. Deane, I. G. Kevrekidis, G. E. Karniadakis and S. A. Orsag, Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinder, Physics of Fluids A, 3 (1991), 2337-2354. doi: 10.1063/1.857881. [13] K. Fukunaga, Introduction to Statistical Recognition, Academic Press, New York, 1990. [14] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin Heidelberg, 1986. doi: 10.1007/978-3-642-61623-5. [15] G. He and Y. N. He, The finite volume method based on stabilized finite element for the stationary Navier-Stokes equations, Journal of Computational and Applied Mathematics, 205 (2007), 651-665. doi: 10.1016/j.cam.2006.07.007. [16] G. He, Y. N. He and X. L. Feng, Finite volume method based on stabilized finite elements for the nonstationary Navier-Stokes problem, Numerical Methods for Partial Differential Equations, 23 (2007), 1167-1191. doi: 10.1002/num.20216. [17] J. G. Heywood and R. Rannacher, Finite element approximation of the non-stationary Navier-Stokes problem, I. Regularity of solutions and second order estimates for spatial discretization, SIAM Journal on Numerical Analysis, 19 (1982), 275-311. doi: 10.1137/0719018. [18] J. G. Heywood and R. Rannacher, Finite element approximation of the non-stationary Navier-Stokes problem part IV: error analysis for second-order time discretization, SIAM Journal on Numerical Analysis, 27 (1990), 353-384. doi: 10.1137/0727022. [19] P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511622700. [20] I. T. Jolliffe, Principal Component Analysis, Springer-Verlag, Berlin, 2002. [21] W. P. Jones and K. R. Menziest, Analysis of the cell-centred finite volume method for the diffusion equation, Journal of Computational Physics, 165 (2000), 45-68. doi: 10.1006/jcph.2000.6595. [22] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90 (2001), 117-148. doi: 10.1007/s002110100282. [23] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40 (2002), 492-515. doi: 10.1137/S0036142900382612. [24] K. Kunisch and S. Volkwein, Control of Burgers's equation by a reduced order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371. doi: 10.1023/A:1021732508059. [25] J. Li and Z. X. Chen, A new stabilized finite volume method for the stationary Stokes equations, Adv. Comput. Math., 30 (2009), 141-152. doi: 10.1007/s10444-007-9060-5. [26] R. H. Li, Z. Y. Chen and W. Wu, Generalized Difference Methods for Differential Equations-Numerical Analysis of Finite Volume Methods, Monographs and Textbooks in Pure and Applied Mathematics 226, Marcel Dekker Inc: New York, 2000. [27] H. Li and Z. D. Luo, A fully-discrete stabilized finite volume element formulation and error analysis based on the Crank-Nicolson method of time discretization for the non-stationary Navier-Stokes equations,, Applied Numerical Mathematics, (). [28] H. R. Li, Z. D. Luo and J. Chen, Numerical simulation based on proper orthogonal decomposition for two-dimensional solute transport problems, Applied Mathematical Modelling, 35 (2011), 2489-2498. doi: 10.1016/j.apm.2010.11.064. [29] H. R. Li, Z. D. Luo and Q. Li, Generalized difference methods for two-dimensional viscoelastic problems, Chinese J. Numer. Math. Appl., 29 (2007), 251-262. doi: 10.1063/1.2744281. [30] H. Li, Z. D. Luo, P. Sun and J. An, A finite volume element formulation and error analysis for the non-stationary conduction-convection problem, Journal of Mathematical Analysis and Applications, 396 (2012), 864-879. doi: 10.1016/j.jmaa.2012.07.046. [31] J. Li, L. H. Shen and Z. X. Chen, Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations, BIT Numerical Mathematics, 50 (2010), 823-842. doi: 10.1007/s10543-010-0277-1. [32] Z. D. Luo, The foundations and Applications of Mixed Finite Element Methods (in Chinese), Chinese Science Press, Beijing, 2006. [33] Z. D. Luo, J. Chen, I. M. Navon and X. Z. Yang, Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations, SIAM Journal on Numerical Analysis, 47 (2008), 1-19. doi: 10.1137/070689498. [34] Z. D. Luo, J. Chen, I. M. Navon and J. Zhu, An optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems, International Journal for Numerical Methods in Fluid, 60 (2009), 409-436. doi: 10.1002/fld.1900. [35] Z. D. Luo, J. Chen, P. Sun and X. Z. Yang, Finite element formulation based on proper orthogonal decomposition for parabolic equations, Science in China Series A: Mathematics, 52 (2009), 585-596. doi: 10.1007/s11425-008-0125-9. [36] Z. D. Luo, J. Chen, Z. H. Xie, J. An and P. Sun, A reduced second-order time accurate finite element formulation based on POD for parabolic equations (in Chinese), Sci. Sin. Math., 41 (2011), 447-460. [37] Z. D. Luo, J. Du, Z. H. Xie and Y. Guo, A reduced stabilized mixed finite element formulation based on proper orthogonal decomposition for the no-stationary Navier-Stokes equations, International Journal for Numerical Methods in Engineering, 88 (2011), 31-46. doi: 10.1002/nme.3161. [38] Z. D. Luo, H. Li, Y. Q. Shang and Z. Fang, A LSMFE formulation based on proper orthogonal decomposition for parabolic equations, Finite Elements in Analysis and Design, 60 (2012), 1-12. doi: 10.1016/j.finel.2012.05.002. [39] Z. D. Luo, H. Li, Y. J. Zhou and X. M. Huang, A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem, Journal of Mathematical Analysis and Applications, 385 (2012), 310-321. doi: 10.1016/j.jmaa.2011.06.057. [40] Z. D. Luo, H. Li, Y. J. Zhou and Z. H. Xie, A reduced finite element formulation and error estimates based on POD method for two-dimensional solute transport problems, Journal of Mathematical Analysis and Applications, 385 (2012), 371-383. doi: 10.1016/j.jmaa.2011.06.051. [41] Z. D. Luo, Z. H. Xie and J. Chen, A reduced MFE formulation based on POD for the non-stationary conduction-convection problems, Acta Mathematica Scientia, 31 (2011), 1765-1785. doi: 10.1016/S0252-9602(11)60360-3. [42] Z. D. Luo, Z. H. Xie, Y. Q. Shang and J. Chen, A reduced finite volume element formulation and numerical simulations based on POD for parabolic equations, Journal of Computational and Applied Mathematics, 235 (2011), 2098-2111. doi: 10.1016/j.cam.2010.10.008. [43] Z. D. Luo, Y. J. Zhou and X. Z. Yang, A reduced finite element formulation based on proper orthogonal decomposition for Burgers equation, Applied Numerical Mathematics, 59 (2009), 1933-1946. doi: 10.1016/j.apnum.2008.12.034. [44] Z. D. Luo, J. Zhu, R. W. Wang and I. M. Navon, Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 4184-4195. doi: 10.1016/j.cma.2007.04.003. [45] H. V. Ly and H. T. Tran, Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor, Quart. Appl. Math., 60 (2002), 631-656. [46] X. Ma and G. E. Karniadakis, A low-dimensional model for simulating three-dimensional cylinder flow, Journal of Fluid Mechanics, 458 (2002), 181-190. doi: 10.1017/S0022112002007991. [47] W. Rudin, Functional and Analysis, $2^{nd}$ edition, McGraw-Hill Companies, Inc., 1973. [48] L. H. Shen, J. Li and Z. X. Chen, Analysis of a stabilized finite volume method for the transient Stokes equations, International Journal of Numerical Analysis and Modeling, 6 (2009), 505-519. [49] E. Süli, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes, SIAM Journal on Numerical Analysis, 28 (1991), 1419-1430. doi: 10.1137/0728073. [50] R. Temam, Navier-Stokes Equations, $3^{rd}$ edition, North-Holland, Amsterdam, New York, 1984. [51] Z. Wang, I. Akhtar, J. Borggaard and T. Iliescu, Two-level discretizations of non-linear closure models for proper orthogonal decomposition, Journal of Computational Physics, 230 (2011), 126-146. doi: 10.1016/j.jcp.2010.09.015. [52] M. Yang and H. L. Song, A postprocessing finite volume method for time-dependent Stokes equations, Applied Numerical Mathematics, 59 (2009), 1922-1932. doi: 10.1016/j.apnum.2009.02.004. [53] X. Ye, On the relation between finite volume and finite element methods applied to the Stokes equations, Numerical Methods for Partial Differential Equations, 17 (2001), 440-453. doi: 10.1002/num.1021.

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References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] D. Ahlman, F. Södelundon, J. Jackson, A. Kurdila and W. Shyy, Proper orthogonal decomposition for time-dependent lid-driven cavity flows, Numer. Heat Trans. Part B-Fund., 42 (2002), 285-306. doi: 10.1080/10407790190053950. [3] I. Ammara and C. Masson, Development of a fully coupled control-volume finite element method for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 44 (2004), 621-644. doi: 10.1002/fld.662. [4] J. An, P. Sun, Z. D. Luo and X. M. Huang, A stabilized fully discrete finite volume element formulation for non-stationary Stokes equations, Math. Numer. Sin., 33 (2011), 213-224. [5] R. E. Bank and D. J. Rose, Some error estimates for the box methods, SIAM Journal on Numerical Analysis, 24 (1987), 777-787. doi: 10.1137/0724050. [6] M. Bergmann, C. H. Bruneau and A. Iollo, Enablers for robust POD models, Journal of Computational Physics, 228 (2009), 516-538. doi: 10.1016/j.jcp.2008.09.024. [7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1. [8] Z. Cai and S. McCormick, On the accuracy of the finite volume element method for diffusion equations on composite grid, SIAM Journal on Numerical Analysis, 27 (1990), 636-655. doi: 10.1137/0727039. [9] J. R. Cannon and Y. Lin, A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM Journal on Numerical Analysis, 27 (1990), 595-607. doi: 10.1137/0727036. [10] S. H. Chou and D. Y. Kwak, A covolume method based on rotated bilinears for the generalized Stokes problem, SIAM Journal on Numerical Analysis, 35 (1998), 494-507. doi: 10.1137/S0036142996299964. [11] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [12] A. E. Deane, I. G. Kevrekidis, G. E. Karniadakis and S. A. Orsag, Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinder, Physics of Fluids A, 3 (1991), 2337-2354. doi: 10.1063/1.857881. [13] K. Fukunaga, Introduction to Statistical Recognition, Academic Press, New York, 1990. [14] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin Heidelberg, 1986. doi: 10.1007/978-3-642-61623-5. [15] G. He and Y. N. He, The finite volume method based on stabilized finite element for the stationary Navier-Stokes equations, Journal of Computational and Applied Mathematics, 205 (2007), 651-665. doi: 10.1016/j.cam.2006.07.007. [16] G. He, Y. N. He and X. L. Feng, Finite volume method based on stabilized finite elements for the nonstationary Navier-Stokes problem, Numerical Methods for Partial Differential Equations, 23 (2007), 1167-1191. doi: 10.1002/num.20216. [17] J. G. Heywood and R. Rannacher, Finite element approximation of the non-stationary Navier-Stokes problem, I. Regularity of solutions and second order estimates for spatial discretization, SIAM Journal on Numerical Analysis, 19 (1982), 275-311. doi: 10.1137/0719018. [18] J. G. Heywood and R. Rannacher, Finite element approximation of the non-stationary Navier-Stokes problem part IV: error analysis for second-order time discretization, SIAM Journal on Numerical Analysis, 27 (1990), 353-384. doi: 10.1137/0727022. [19] P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511622700. [20] I. T. Jolliffe, Principal Component Analysis, Springer-Verlag, Berlin, 2002. [21] W. P. Jones and K. R. Menziest, Analysis of the cell-centred finite volume method for the diffusion equation, Journal of Computational Physics, 165 (2000), 45-68. doi: 10.1006/jcph.2000.6595. [22] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90 (2001), 117-148. doi: 10.1007/s002110100282. [23] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40 (2002), 492-515. doi: 10.1137/S0036142900382612. [24] K. Kunisch and S. Volkwein, Control of Burgers's equation by a reduced order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371. doi: 10.1023/A:1021732508059. [25] J. Li and Z. X. Chen, A new stabilized finite volume method for the stationary Stokes equations, Adv. Comput. Math., 30 (2009), 141-152. doi: 10.1007/s10444-007-9060-5. [26] R. H. Li, Z. Y. Chen and W. Wu, Generalized Difference Methods for Differential Equations-Numerical Analysis of Finite Volume Methods, Monographs and Textbooks in Pure and Applied Mathematics 226, Marcel Dekker Inc: New York, 2000. [27] H. Li and Z. D. Luo, A fully-discrete stabilized finite volume element formulation and error analysis based on the Crank-Nicolson method of time discretization for the non-stationary Navier-Stokes equations,, Applied Numerical Mathematics, (). [28] H. R. Li, Z. D. Luo and J. Chen, Numerical simulation based on proper orthogonal decomposition for two-dimensional solute transport problems, Applied Mathematical Modelling, 35 (2011), 2489-2498. doi: 10.1016/j.apm.2010.11.064. [29] H. R. Li, Z. D. Luo and Q. Li, Generalized difference methods for two-dimensional viscoelastic problems, Chinese J. Numer. Math. Appl., 29 (2007), 251-262. doi: 10.1063/1.2744281. [30] H. Li, Z. D. Luo, P. Sun and J. An, A finite volume element formulation and error analysis for the non-stationary conduction-convection problem, Journal of Mathematical Analysis and Applications, 396 (2012), 864-879. doi: 10.1016/j.jmaa.2012.07.046. [31] J. Li, L. H. Shen and Z. X. Chen, Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations, BIT Numerical Mathematics, 50 (2010), 823-842. doi: 10.1007/s10543-010-0277-1. [32] Z. D. Luo, The foundations and Applications of Mixed Finite Element Methods (in Chinese), Chinese Science Press, Beijing, 2006. [33] Z. D. Luo, J. Chen, I. M. Navon and X. Z. Yang, Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations, SIAM Journal on Numerical Analysis, 47 (2008), 1-19. doi: 10.1137/070689498. [34] Z. D. Luo, J. Chen, I. M. Navon and J. Zhu, An optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems, International Journal for Numerical Methods in Fluid, 60 (2009), 409-436. doi: 10.1002/fld.1900. [35] Z. D. Luo, J. Chen, P. Sun and X. Z. Yang, Finite element formulation based on proper orthogonal decomposition for parabolic equations, Science in China Series A: Mathematics, 52 (2009), 585-596. doi: 10.1007/s11425-008-0125-9. [36] Z. D. Luo, J. Chen, Z. H. Xie, J. An and P. Sun, A reduced second-order time accurate finite element formulation based on POD for parabolic equations (in Chinese), Sci. Sin. Math., 41 (2011), 447-460. [37] Z. D. Luo, J. Du, Z. H. Xie and Y. Guo, A reduced stabilized mixed finite element formulation based on proper orthogonal decomposition for the no-stationary Navier-Stokes equations, International Journal for Numerical Methods in Engineering, 88 (2011), 31-46. doi: 10.1002/nme.3161. [38] Z. D. Luo, H. Li, Y. Q. Shang and Z. Fang, A LSMFE formulation based on proper orthogonal decomposition for parabolic equations, Finite Elements in Analysis and Design, 60 (2012), 1-12. doi: 10.1016/j.finel.2012.05.002. [39] Z. D. Luo, H. Li, Y. J. Zhou and X. M. Huang, A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem, Journal of Mathematical Analysis and Applications, 385 (2012), 310-321. doi: 10.1016/j.jmaa.2011.06.057. [40] Z. D. Luo, H. Li, Y. J. Zhou and Z. H. Xie, A reduced finite element formulation and error estimates based on POD method for two-dimensional solute transport problems, Journal of Mathematical Analysis and Applications, 385 (2012), 371-383. doi: 10.1016/j.jmaa.2011.06.051. [41] Z. D. Luo, Z. H. Xie and J. Chen, A reduced MFE formulation based on POD for the non-stationary conduction-convection problems, Acta Mathematica Scientia, 31 (2011), 1765-1785. doi: 10.1016/S0252-9602(11)60360-3. [42] Z. D. Luo, Z. H. Xie, Y. Q. Shang and J. Chen, A reduced finite volume element formulation and numerical simulations based on POD for parabolic equations, Journal of Computational and Applied Mathematics, 235 (2011), 2098-2111. doi: 10.1016/j.cam.2010.10.008. [43] Z. D. Luo, Y. J. Zhou and X. Z. Yang, A reduced finite element formulation based on proper orthogonal decomposition for Burgers equation, Applied Numerical Mathematics, 59 (2009), 1933-1946. doi: 10.1016/j.apnum.2008.12.034. [44] Z. D. Luo, J. Zhu, R. W. Wang and I. M. Navon, Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 4184-4195. doi: 10.1016/j.cma.2007.04.003. [45] H. V. Ly and H. T. Tran, Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor, Quart. Appl. Math., 60 (2002), 631-656. [46] X. Ma and G. E. Karniadakis, A low-dimensional model for simulating three-dimensional cylinder flow, Journal of Fluid Mechanics, 458 (2002), 181-190. doi: 10.1017/S0022112002007991. [47] W. Rudin, Functional and Analysis, $2^{nd}$ edition, McGraw-Hill Companies, Inc., 1973. [48] L. H. Shen, J. Li and Z. X. Chen, Analysis of a stabilized finite volume method for the transient Stokes equations, International Journal of Numerical Analysis and Modeling, 6 (2009), 505-519. [49] E. Süli, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes, SIAM Journal on Numerical Analysis, 28 (1991), 1419-1430. doi: 10.1137/0728073. [50] R. Temam, Navier-Stokes Equations, $3^{rd}$ edition, North-Holland, Amsterdam, New York, 1984. [51] Z. Wang, I. Akhtar, J. Borggaard and T. Iliescu, Two-level discretizations of non-linear closure models for proper orthogonal decomposition, Journal of Computational Physics, 230 (2011), 126-146. doi: 10.1016/j.jcp.2010.09.015. [52] M. Yang and H. L. Song, A postprocessing finite volume method for time-dependent Stokes equations, Applied Numerical Mathematics, 59 (2009), 1922-1932. doi: 10.1016/j.apnum.2009.02.004. [53] X. Ye, On the relation between finite volume and finite element methods applied to the Stokes equations, Numerical Methods for Partial Differential Equations, 17 (2001), 440-453. doi: 10.1002/num.1021.
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