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Characteristic Local Discontinuous Galerkin Methods for Incompressible Navier-Stokes Equations

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  03 May 2017

Shuqin Wang ,
Weihua Deng ,
Jinyun Yuan  and
Yujiang Wu
Shuqin Wang*
Affiliation:
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China Department of Mathematics, Federal University of Paraná, Centro Politécnico, CP: 19.081, Curitiba, CEP: 81531-990, PR, Brazil
Weihua Deng*
Affiliation:
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China
Jinyun Yuan*
Affiliation:
Department of Mathematics, Federal University of Paraná, Centro Politécnico, CP: 19.081, Curitiba, CEP: 81531-990, PR, Brazil
Yujiang Wu*
Affiliation:
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China
*
*Corresponding author. Email addresses:[email protected] (S. Wang), [email protected] (W. Deng), [email protected] (J. Yuan), [email protected] (Y.Wu)
*Corresponding author. Email addresses:[email protected] (S. Wang), [email protected] (W. Deng), [email protected] (J. Yuan), [email protected] (Y.Wu)
*Corresponding author. Email addresses:[email protected] (S. Wang), [email protected] (W. Deng), [email protected] (J. Yuan), [email protected] (Y.Wu)
*Corresponding author. Email addresses:[email protected] (S. Wang), [email protected] (W. Deng), [email protected] (J. Yuan), [email protected] (Y.Wu)
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Abstract

By combining the characteristic method and the local discontinuous Galerkin method with carefully constructing numerical fluxes, variational formulations are established for time-dependent incompressible Navier-Stokes equations in ℝ2. The nonlinear stability is proved for the proposed symmetric variational formulation. Moreover, for general triangulations the priori estimates for the L2–norm of the errors in both velocity and pressure are derived. Some numerical experiments are performed to verify theoretical results.

Keywords

Navier-Stokes equationslocal discontinuous Galerkin methodsymmetric variational formulation

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M15: Error bounds 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 1 , July 2017 , pp. 202 - 227
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

Communicated by Chi-Wang Shu

References

[1] Achdou, Y., Guermond, J.-L., Convergence analysis of a finite element projection/lagrange-Galerkin method for the incompressible Navier-Stokes equations, SIAM J. Numer. Anal. 37 (2000), 799–826.Google Scholar
[2] Arbogast, T., Wheeler, M. F., A characteristics-mixed finite element method for advection-dominated transport problems, SIAM J. Numer. Anal., 32 (1995), 404–424.CrossRefGoogle Scholar
[3] Bassi, F., Rebay, S., A high-order accurate discontinuous finite elemnet method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), 267–279.CrossRefGoogle Scholar
[4] Boukir, K., Maday, Y., Métivet, B., A high order characteristics/ finite element method for the incompressible Navier-Stokes equations, Int. J. Comput. Numer. Methods Fluids, 25 (1997), 1421–1454.Google Scholar
[5] Castillo, P., Cockburn, B., Perugia, I., and Schötzau, D., An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 38 (2000), 1676–1706.Google Scholar
[6] Chen, Z. X., Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems, Comput. Meth. Appl. Mech. Engrg., 191 (2002), 2509–2538.Google Scholar
[7] Chen, Z. X., Ewing, R. E., Jiang, Q. Y., and Spagnuolo, A. M., Error analysis for characteristics-based methods for degenerate parabolic problems, SIAM J. Numer. Anal., 40 (2002), 1491–1515.Google Scholar
[8] Chorin, A., Marsden, J. E., A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, New York, 2000.Google Scholar
[9] Chrysafinos, K., Walkington, N. J., Discontinuous Galerkin approximations of the Stokes and Navier-Stokes equations, Math. Comp., 79 (2010), 2135–2167.CrossRefGoogle Scholar
[10] Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C., Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal., 40 (2003), 319–343.CrossRefGoogle Scholar
[11] Cockburn, B., Kanschat, G., Schötzau, D., A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp., 74 (2005), 1067–1095.Google Scholar
[12] Cockburn, B., Kanschat, G., Schötzau, D., A note on discontinuous Galerkin divergence free solutions of Navier-Stokes equations, J. Sci. Comput., 31 (2007), 61–73.CrossRefGoogle Scholar
[13] Cockburn, B., Shu, C. -W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440–2463.Google Scholar
[14] Cockburn, B., Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection dominated problems, J. Sci. Comput., 16 (2001), 173–261.Google Scholar
[15] Douglas, J. Jr. and Russell, T. F., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871–885.Google Scholar
[16] Girault, V., Raviart, P., Finite Element Approximations for the Navier-Stokes Equations, Springer-Verlag, New York, 1986.CrossRefGoogle Scholar
[17] Girault, V., Rivière, B., Wheeler, M. F., A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations, ESAIM Math. Model. Numer. Anal., 39 (2005), 1115–1147.CrossRefGoogle Scholar
[18] Guermond, J. L., Shen, J., On the error estimates for the rotational pressure correction projection method, Math. Comp., 73 (2003), 1719–1737.CrossRefGoogle Scholar
[19] He, Y., A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem, IMA J. Numer. Anal., 23 (2003), 665–691.CrossRefGoogle Scholar
[20] Hesthaven, J. S., Warburton, T., Nodal discontinuous Galerkin methods: Algorithms, Analysis, and Applications. Springer-Verlag, New York, USA, 2008.Google Scholar
[21] Liu, M. E., Ren, Y. X., Zhang, H. X., A class of fully second order accurate projection methods for solving the incompressible Navier-Stokes equations, J. Comput. Phys., 200 (2004), 325–346.CrossRefGoogle Scholar
[22] Nochetto, R. H., Pyo, J. -H., The Gauge-Uzawa finite element method. Part I: The Navier-Stokes Equations, SIAM J. Numer. Anal., 43 (2005), 1043–1086.Google Scholar
[23] Rivière, B., Discontinuous Galerkin methods for solving elliptic and parabolic equations: Theory and Implementation, Society for Industrial and Applied Mathematics, SIAM, 1999.Google Scholar
[24] Rivière, B., Girault, V., Discontinuous finite element methods for incompressible flows on subdomains with non-matching interfaces, Comput. Meth. Appl. Mech. Engrg., 195 (2006), 3274–3292.Google Scholar
[25] Si, Z. Y., Song, X. G., Huang, P. Z., Modified characteristics Gauge-Uzawa finite element method for time dependent conduction-convection problems, J. Sci. Comput., 58 (2014), 1–24.Google Scholar
[26] Süli, E., Convergence and nonlinear stability of Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math., 53 (1988), 459–483.Google Scholar
[27] Temam, R., Navier-Stokes Equations: Theory and numerical analysis, North-Holland-Amsterdam. New York. Oxford, Elsevier Science Publishers B.V., 1984.Google Scholar
[28] Wang, H., Wang, K. X., Uniform estimates for Eulerian-Lagrangian methods for singularly perturbed time-dependent problems, SIAM J. Numer. Anal., 45 (2007), 1305–1329.Google Scholar
[29] Wang, H., Wang, K. X., Uniform estimates of an Eulerian-Lagrangian method for time-dependent convection-diffusion equations in multiple space dimensions, SIAM J. Numer. Anal., 48 (2010), 1444-1473.Google Scholar
[30] Wang, H., An optimal-order error estimate for a family of ELLAM-MFEM approximations to porous medium flow, SIAM J. Numer. Anal., 46 (2008), 2133–2152.Google Scholar
[31] Wang, K. X., Wang, H., Al-Lawatia, M., Rui, H. X., A family of characteristic discontinuous Galerkinmethods for transient, advection-diffusion equations and their optimal-order L 2 error estimates, Commun. Comput. Phys., 6 (2009), 203–230.Google Scholar
[32] Wang, K. X., Wang, H., Uniform estimates for a family of Eulerian-Lagrangian methods for time-dependent convection-diffusion equations with degenerate diffusion, IMA J. Numer. Anal., 31 (2011), 1006–1037.Google Scholar