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Characteristic stabilized finite element method for the transient Navier-Stokes equations. (English) Zbl 1231.76150

Summary: Based on the lowest equal-order conforming finite element subspace \((X_{h}, M_{h})\) (i.e. \(P_{1}-P_{1}\) or \(Q_{1}- Q_{1}\) elements), a characteristic stabilized finite element method for transient Navier-Stokes problem is proposed. The proposed method has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, and averting the difficulties caused by trilinear terms. Existence,uniqueness and error estimates of the approximate solution are proved by applying the technique of characteristic finite element method. Finally, a serious of numerical experiments are given to show that this method is highly efficient for transient Navier-Stokes problem.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Douglas, J.; Thomas, F.; Russell, Numerical method for convection-dominated diffusion problem based on combining the method of characteristics with finite element of finite difference procedures, SIAM J. Numer. Anal., 19, 5, 871-885 (1982) · Zbl 0492.65051
[2] Pironneau, O., On the transport-diffusion algorithm and its application to the Navier-Stokes equations, Numer. Math., 38, 309-332 (1982) · Zbl 0505.76100
[3] Süli, Endre., Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math., 53, 459-483 (1998) · Zbl 0637.76024
[4] Boukir, K.; Maday, Y., A high-order characteristics/finite element method for the incompressible Navier-Stokes equations, Int. J. Numer. Method Fluids, 25, 1421-1454 (1997) · Zbl 0904.76040
[5] Alejandro, Allievi; Rodolfo, Bermejo, Finite element modified of characteristics for the Navier-Stokes equations, Int. J. Numer. Method Fluids, 32, 439-464 (2000) · Zbl 0955.76048
[6] Achdou, Y.; Guermond, J. L., Analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 37, 3, 799-826 (2000) · Zbl 0966.76041
[7] Benqué, J. P.; Labdie, G.; Ronat, J., A new finite element method for Navier-Stokes equations coupled with a temperature equation, (Kawai, T., Proceedings of 4th International Symposium on Finite Element Methods in Flow Problem (1982), North-Holland: North-Holland Amsterdam), 295-302 · Zbl 0508.76049
[8] Bercovier, M.; Pironneau, O., Characteristics and finite element method, (Kawai, T., Proceedings of 4th Internatioal Symposium On Finite Element Methods in Flow Problem (1982), North-Holland: North-Holland Amsterdam) · Zbl 0508.76007
[9] Hasbani, Y.; Livne, E.; Bercovier, M., Finite element and characteristics applied to advection-diffusion equation, Comput. Fluids, 11, 71-83 (1983) · Zbl 0511.76089
[12] Smith, B.; Bjorstad, P.; Grropp, W., Domain Decomposition. Domain Decomposition, Parallel Multilevel Method for Elliptic Partial Differential Equations (1996), Cambridge University Press: Cambridge University Press Cambridge
[13] Brooks, A.; Hughes, T., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 1-3, 199-259 (1982) · Zbl 0497.76041
[14] Hughes, T.; Franca, L.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. Circumventing the BabuskaCBrezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59, 1, 85-99 (1986) · Zbl 0622.76077
[16] Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. Comput., 52, 495-508 (1989) · Zbl 0669.76051
[17] Franca, L.; Frey, F., Stabilized finite element methods: II. The incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 99, 2-3, 209-233 (1992) · Zbl 0765.76048
[18] Franca, L.; Hughes, T., Convergence analyses of Galerkin-least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 105, 2, 285-298 (1993) · Zbl 0771.76037
[19] Franca, L.; Stenberg, R., Error analysis of some Galerkin least squares methods for the elasticity equations, SIAM J. Numer. Anal., 28, 6, 1680-1697 (1991) · Zbl 0759.73055
[20] Baiocchi, C.; Brezzi, F.; Franca, L., Virtual bubble and Galerkin-least-squares type methods (Ga.L.S.), Comput. Methods Appl. Mech. Engrg., 105, 1, 125-141 (1993) · Zbl 0772.76033
[21] Brezzi, F.; Bristeau, M.; Franca, L.; Mallet, M.; Roge, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. Methods Appl. Mech. Engrg., 96, 1, 117-129 (1992) · Zbl 0756.76044
[22] Barrenechea, G. R.; Valentin, F., An unusual stabilized finite element method for a generalized Stokes problem, Numer. Math., 92, 4, 653-677 (2002) · Zbl 1019.65087
[23] Codina, R.; Blasco, J.; Buscaglia, G.; Huerta, A., Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection, Int. J. Numer. Methods Fluids, 37, 4, 419-444 (2001) · Zbl 1074.76032
[24] Becker, R.; Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo, 38, 4, 173-199 (2001) · Zbl 1008.76036
[25] Silvester, D. J., Optimal low-order finite element methods for incompressible flow, Comput. Methods Appl. Mech. Engrg., 111, 3-4, 357-368 (1994) · Zbl 0844.76059
[27] Bochev, Pavel B.; Dohrmann, Clark R.; Gunzburger, Max D., Stabilized of low-order mixed finite element for the stokes equations, SIAM J. Numer. Anal., 44, 1, 82-101 (2006) · Zbl 1145.76015
[28] Li, J.; He, Y.; Chen, Z., A new stabilized finite element method for the transient Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 197, 1, 22-35 (2007) · Zbl 1169.76392
[29] Dohrmann, C.; Bochev, P., A stabilized finite element method for the Stokes problem based on polynomial pressure projections, Int. J. Numer. Methods Fluids, 46, 2, 183-201 (2004) · Zbl 1060.76569
[30] Li, J.; He, Y., A stabilized finite element method based on two local Gauss integrations for the Stokes equations, J. Comput. Appl. Math., 214, 1, 58-65 (2008) · Zbl 1132.35436
[31] He, Y.; Li, J., A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations, Appl. Numer. Math., 58, 10, 1503-1514 (2008) · Zbl 1155.35406
[32] Shang, Yueqiang, New stabilized finite element method for time-dependent incompressible flow problems, Int. J. Numer. Method Fluids, 62, 166-187 (2010) · Zbl 1422.76048
[33] He, Y., A fully discrete stabilized finite-element method for the timedependent Navier-Stokes problem, IMA J. Numer. Anal., 23, 665-691 (2003) · Zbl 1135.76331
[34] He, Y.; Sun, W., Stabilized finite element method based on the Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations, Math. Comput., 76, 257, 115-136 (2007) · Zbl 1129.35004
[35] Chen, Yumei; Luo, Yan; Feng, Minfu, A stabilized characteristic finite-element methods for the non-stationary Navier-Stokes equation, Numer. Math.: J. Chin. Univ., 29, 4, 350-357 (2007) · Zbl 1150.76436
[36] Brefort, B.; Ghidaglia, J. M.; Temam, R., Attractor for the penalty Navier-Stokes equations, SIAM J. Math. Anal, 19, 1-21 (1988) · Zbl 0696.35131
[37] Girault, V.; Raviart, P. A., Finite Element Method for Navier-Stokes Equations: Theory and Algorithms (1987), Springer-Verlag: Springer-Verlag Berlin and Heidelberg · Zbl 0396.65070
[38] Girault, V.; Raviart, P., Finite Element Methods for Navier-Stokes Equations (1986), Springer: Springer Berlin · Zbl 0585.65077
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