×

Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. (English) Zbl 1015.76045

Summary: We present a stabilized finite element method to solve the transient Navier-Stokes equations based on the decomposition of unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and to use equal velocity-pressure interpolations. Three main issues are addressed. The first is to estimate the behavior of stabilization parameters based on Fourier analysis of the problem for the subscales. Secondly, we discuss the way to deal with transient problems discretized using a finite difference scheme. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Simo, J. C.; Armero, F., Unconditional stability and long term behavior of transient algorithms for the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 111, 111-154 (1994) · Zbl 0846.76075
[2] Blasco, J.; Codina, R., Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier-Stokes equations, Applied Numerical Mathematics, 38, 475-497 (2001) · Zbl 1011.76041
[3] Temam, R., Navier-Stokes equations (1984), North-Holland: North-Holland Amsterdam · Zbl 0572.35083
[4] Girault, V.; Raviart, P. A., Finite Element Approximation for Navier-Stokes Equations, (Lecture Notes in Mathematics, vol. 749 (1979), Springer: Springer Berlin) · Zbl 0396.65070
[5] Hughes, T. J.R., Multiscale phenomena: Green’s function, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized formulations, Computer Methods in Applied Mechanics and Engineering, 127, 387-401 (1995) · Zbl 0866.76044
[6] Hughes, T. J.R.; Feijóo, G. R.; Mazzei, L.; Quincy, J. B., The variational multiscale method-a paradigm for computational mechanics, Computer Methods in Applied Mechanics and Engineering, 166, 3-24 (1998) · Zbl 1017.65525
[7] Baiocchi, C.; Brezzi, F.; Franca, L. P., Virtual bubbles and Galerkin/least-squares type methods (Ga.L.S), Computer Methods in Applied Mechanics and Engineering, 105, 125-141 (1993) · Zbl 0772.76033
[8] Brezzi, F.; Franca, L. P.; Hughes, T. J.R.; Russo, A., \(b\)=∫\(g\), Computer Methods in Applied Mechanics and Engineering, 145, 329-339 (1997) · Zbl 0904.76041
[9] Codina, R., Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods, Computer Methods in Applied Mechanics and Engineering, 190, 1579-1599 (2000) · Zbl 0998.76047
[10] Codina, R.; Blasco, J., A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation, Computer Methods in Applied Mechanics and Engineering, 143, 373-391 (1997) · Zbl 0893.76040
[11] Codina, R.; Blasco, J., Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations, Numerische Mathematik, 87, 59-81 (2000) · Zbl 0988.76049
[12] Guermond, J. L., Stabilization of Galerkin approximations of transport equations by subgrid modeling, Mathematical Modelling and Numerical Analysis, 33, 1293-1316 (1999) · Zbl 0946.65112
[13] Hughes, T. J.R.; Mazzei, L.; Jansen, K. E., Large eddy simulation and the variational multiscale method, Computing and Visualization in Science, 3, 47-59 (2000) · Zbl 0998.76040
[14] Brezzi, F.; Bristeau, M. O.; Franca, L.; Mallet, M.; Rogé, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Computer Methods in Applied Mechanics and Engineering, 96, 117-129 (1992) · Zbl 0756.76044
[15] Brezzi, F.; Marini, D.; Süli, E., Residual-free bubbles for advection-diffusion problems: the general error analysis, Numerische Mathematik, 85, 31-47 (2000) · Zbl 0963.65109
[16] Canuto, C.; Van Kemenade, V., Bubble-stabilized spectral methods for the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 135, 35-61 (1996) · Zbl 0894.76057
[17] Franca, L. P.; Nesliturk, A.; Stynes, M., On the stability of residual free bubbles for convection-diffusion problems and their approximation by a two-level finite element method, Computer Methods in Applied Mechanics and Engineering, 166, 35-49 (1998) · Zbl 0934.65127
[18] Russo, A., Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 132, 335-343 (1996) · Zbl 0887.76038
[19] Pierre, R., Optimal selection of the bubble function in the stabilization of the Pl-Pl element for the Stokes problem, SIAM Journal on Numerical Analysis, 32, 1210-1224 (1995) · Zbl 0833.76037
[20] Ammi, A. A.O.; Marion, M., Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations, Numerische Mathematik, 68, 189-213 (1994) · Zbl 0811.76035
[21] Brown, R. M.; Perry, P.; Shen, Z., The additive turbulent decomposition for the two-dimensional incompressible Navier-Stokes equations: convergence theorems and error estimates, SIAM Journal on Numerical Analysis, 59, 139-155 (1998) · Zbl 0922.35123
[22] Burie, J. B.; Marion, M., Multilevel methods in space and time for the Navier-Stokes equations, SIAM Journal on Numerical Analysis, 34, 1574-1599 (1997) · Zbl 0897.76070
[23] Foias, C.; Manley, O.; Temam, R., Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, Mathematical Modelling and Numerical Analysis, 22, 93-118 (1988) · Zbl 0663.76054
[24] Jauberteau, F.; Rosier, C.; Temam, R., A nonlinear Galerkin method for the Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 80, 245-260 (1990) · Zbl 0722.76039
[25] Marion, M.; Temam, R., Nonlinear Galerkin methods, SIAM Journal on Numerical Analysis, 26, 157-1139 (1989) · Zbl 0683.65083
[26] R. Codina, Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales, Numerische Mathematik, submitted for publication; R. Codina, Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales, Numerische Mathematik, submitted for publication · Zbl 1144.76029
[27] Chacón, T., A term by term stabilization algorithm for the finite element solution of incompressible flow problems, Numerische Mathematik, 79, 283-319 (1998) · Zbl 0910.76033
[28] Franca, L. P.; Hughes, T. J.R., Convergence analyses of Galerkin least-squares methods for advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 105, 285-298 (1993) · Zbl 0771.76037
[29] Franca, L. P.; Frey, S. L., Stabilized finite element methods: II. The incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 99, 209-233 (1992) · Zbl 0765.76048
[30] Roos, H.-G.; Stynes, M.; Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations-Convection-Diffusion and Flow Problems (1996), Springer: Springer Berlin · Zbl 0844.65075
[31] Tobiska, L.; Lube, G., A modified streamline-diffusion method for solving the stationary Navier-Stokes equations, Numerische Mathematik, 59, 13-29 (1991) · Zbl 0696.76034
[32] Tobiska, L.; Verfürth, R., Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations, SIAM Journal on Numerical Analysis, 33, 107-127 (1996) · Zbl 0843.76052
[33] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer: Springer Berlin · Zbl 0788.73002
[34] Codina, R., Comparison of some finite element methods for solving the diffusion-convection-reaction equation, Computer Methods in Applied Mechanics and Engineering, 156, 185-210 (1998) · Zbl 0959.76040
[35] Codina, R., A stabilized finite element method for generalized stationary incompressible flows, Computer Methods in Applied Mechanics and Engineering, 190, 2681-2706 (2001) · Zbl 0996.76045
[36] Tezduyar, T. E.; Mittal, S.; Ray, S. E.; Shih, R., Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Computer Methods in Applied Mechanics and Engineering, 95, 221-242 (1992) · Zbl 0756.76048
[37] Gresho, P. M., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part I: Theory, International Journal for Numerical Methods in Fluids, 11, 587-620 (1990) · Zbl 0712.76035
[38] Shakib, F.; Hughes, T. J.R., A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least-squares algorithms, Computer Methods in Applied Mechanics and Engineering, 87, 35-58 (1991) · Zbl 0760.76051
[39] Codina, R.; Blasco, J.; Buscaglia, G. C.; Huerta, A., Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection, International Journal for Numerical Methods in Fluids, 37, 419-444 (2001) · Zbl 1074.76032
[40] O. Soto, R. Codina, A numerical model for mould filling using a stabilized finite element method and the VOF technique, International Journal for Numerical Methods in Fluids, submitted for publication; O. Soto, R. Codina, A numerical model for mould filling using a stabilized finite element method and the VOF technique, International Journal for Numerical Methods in Fluids, submitted for publication · Zbl 1010.76053
[41] Jansen, K. E.; Collis, S. S.; Whiting, C.; Shakib, F., A better consistency for low-order stabilized finite element methods, Computer Methods in Applied Mechanics and Engineering, 174, 153-170 (1999) · Zbl 0956.76044
[42] Lube, G.; Weiss, D., Stabilized finite element methods for singularly perturbed parabolic problems, Applied Numerical Mathematics, 17, 431-459 (1995) · Zbl 0838.65095
[43] Codina, R., A nodal-based implementation of a stabilized finite element method for incompressible flow problems, International Journal for Numerical Methods in Fluids, 33, 737-766 (2000) · Zbl 0987.76048
[44] Ghia, U.; Ghia, K. N.; Shin, C. T., High-resolutions for incompressible flow using the Navier-Stokes equations and a multi-grid method, Journal of Computational Physics, 48, 387-441 (1982) · Zbl 0511.76031
[45] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation, Computer Methods in Applied Mechanics and Engineering, 32, 199-259 (1982) · Zbl 0497.76041
[46] Gresho, P. M.; Chan, S. T.; Lee, R. L.; Upson, C. D., A modified finite element method for solving the time-dependent, incompressible Navier-Stokes equations. Part 2: Applications, International Journal for Numerical Methods in Fluids, 4, 619-640 (1984) · Zbl 0559.76031
[47] Engelman, M. S.; Jamnia, M. A., Transient flow past a circular cylinder: a benchmark solution, International Journal for Numerical Methods in Fluids, 11, 985-1000 (1990)
[48] Behr, M. A.; Hastreiter, D.; Mittal, S.; Tezduyar, T. E., Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries, Computer Methods in Applied Mechanics and Engineering, 123, 309-316 (1995)
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.