## A multiscale finite element method for the incompressible Navier-Stokes equations.(English)Zbl 1178.76233

Summary: A new multiscale finite element method for the incompressible Navier-Stokes equations. The proposed method arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. Modeling of the unresolved scales corrects the lack of stability of the standard Galerkin formulation and yields a method that possesses superior properties like that of the streamline upwind/Petrov-Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The multiscale method allows arbitrary combinations of interpolation functions for the velocity and the pressure fields, specifically the equal order interpolations that are easy to implement but violate the celebrated Babuska-Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor $$\tau$$ appears naturally via the solution of the fine-scale problem. A family of 2-D elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed. Convergence studies for the method on uniform, skewed as well as composite meshes are presented. Numerical simulations of the nonlinear steady and transient flow problems are shown that exhibit the good stability and accuracy properties of the method.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids
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### References:

 [1] Ayub, M.; Masud, A., A new stabilized formulation for convective-diffusive heat transfer, Numer. heat transfer, 43, 6, 601-625, (2003) [2] Baiocchi, C.; Brezzi, F.; Franca, L.P., Virtual bubbles and Galerkin-least-squares type methods (ga.L.S), Comput. methods appl. mech. engrg., 105, 125-141, (1993) · Zbl 0772.76033 [3] Behr, M.A.; Franca, L.P.; Tezduyar, T.E., Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows, Comput. methods appl. mech. engrg., 104, 31-48, (1993) · Zbl 0771.76033 [4] Brezzi, F.; Bristeau, M.O.; Franca, L.P.; 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 [5] Brezzi, F.; Franca, L.P.; Hughes, T.J.R.; Russo, A., b=∫g, Comput. methods appl. mech. engrg., 145, 3-4, 329-339, (1997) · Zbl 0904.76041 [6] Brezzi, F.; Franca, L.P.; Russo, A., Further considerations on residual-free bubbles for advective-diffusive equations, Comput. methods appl. mech. engrg., 166, 1-2, 25-33, (1998) · Zbl 0934.65126 [7] Brezzi, F.; Houston, P.; Marini, D.; Suli, E., Modeling subgrid viscosity for advection-diffusion problems, Comput. methods appl. mech. engrg., 190, 1601-1610, (2000) · Zbl 0980.76042 [8] Brezzi, F.; Marini, D.; Russo, A., Application of the pseudo residual-free bubbles to the stabilization of convection-diffusion problems, Comput. methods appl. mech. engrg., 166, 51-63, (1998) · Zbl 0932.65113 [9] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041 [10] Codina, R.; Soto, O., Approximation of the incompressible Navier-Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes, Comput. methods appl. mech. engrg., 193, 1403-1419, (2004) · Zbl 1079.76579 [11] Franca, L.P.; Farhat, C., Bubble functions prompt unusual stabilized finite element methods, Comput. methods appl. mech. engrg., 123, 1-4, 299-308, (1995) · Zbl 1067.76567 [12] Franca, L.P.; Farhat, C.; Lesoinne, M.; Russo, A., Unusual stabilized finite element methods and residual free bubbles, Int. J. numer. methods fluids, 27, 2, 159-168, (1998) · Zbl 0904.76045 [13] Franca, L.P.; Frey, S.L., Stabilized finite element methods. II. the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 99, 209-233, (1992) · Zbl 0765.76048 [14] Franca, L.P.; Nesliturk, A., On a two-level finite element method for the incompressible Navier-Stokes equations, Int. J. numer. methods engrg., 52, 433-453, (2001) · Zbl 1002.76066 [15] Ghia, U.; Ghia, K.N.; Shin, C.T., High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. comp. phys., 48, 387-411, (1982) · Zbl 0511.76031 [16] Gravemeier, V.; Wall, W.A.; Ramm, E., A three-level finite element method for the instationary incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 193, 1323-1366, (2004) · Zbl 1085.76038 [17] Grove, A.S.; Shair, F.H.; Petersen, E.E., An experimental investigation of the steady separated flow past a circular cyliner, J. fluid mech., 19, 60, (1964) · Zbl 0117.42506 [18] Hansbo, P.; Szepessy, A., A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 84, 175-192, (1990) · Zbl 0716.76048 [19] Hauke, G.; Hughes, T.J.R., A comparative study of different sets of variables for solving compressible and incompressibe flows, Comput. methods appl. mech. engrg., 153, 1-44, (1998) · Zbl 0957.76028 [20] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044 [21] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewoods Cliffs, NJ, Dover edition, 2000 [22] Hughes, T.J.R.; Feijoo, G.R.; Luca, M.; Jean-Baptiste, Q., The variational multiscale method-a paradigm for computational mechanics, Comput. methods appl. mech. engrg., 166, 1-2, 3-24, (1998) · Zbl 1017.65525 [23] Hughes, T.J.R.; Franca, L.P.; Balestra, M., A new finite element formulation for computational fluid dynamics. V. circumventing the babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. methods appl. mech. engrg., 59, 85-99, (1986) · Zbl 0622.76077 [24] Hughes, T.J.R.; Franca, L.P.; Hulbert, G.M., A new finite element formulation for computational fluid dynamics. VIII. the Galerkin/least-squares method for advective-diffusive equations, Comput. methods appl. mech. engrg., 73, 2, 173-189, (1989) · Zbl 0697.76100 [25] Hughes, T.J.R.; Tezduyar, T.E., Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput. methods appl. mech. engrg., 45, 217-284, (1984) · Zbl 0542.76093 [26] Johnson, C.; Saranen, J., Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. comp., 47, 1-18, (1986) · Zbl 0609.76020 [27] Koobus, B.; Farhat, C., A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes—application to vortex shedding, Comput. methods appl. mech. engrg., 193, 1367-1383, (2004) · Zbl 1079.76567 [28] A. Masud, On a stabilized finite element formulation for incompressible Navier-Stokes equations, in: Proceedings of the Fourth US-Japan Conference on Computational Fluid Dynamics, Tokyo, Japan, May 28-30, 2002. [29] Masud, A.; Bergman, L.A., Application of multiscale finite element methods to the solution of the Fokker-Planck equation, Comput. methods appl. mech. engrg., 194, 1513-1526, (2005) · Zbl 1136.82359 [30] Masud, A.; Hughes, T.J.R., A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems, Comput. methods appl. mech. engrg., 146, 91-126, (1997) · Zbl 0899.76259 [31] Masud, A.; Hughes, T.J.R., A stabilized mixed finite element method for Darcy flow, Comput. methods appl. mech. engrg., 191, 4341-4370, (2002) · Zbl 1015.76047 [32] Masud, A.; Khurram, R.A., A multiscale/stabilized finite element method for the advection-diffusion equation, Comput. methods appl. mech. engrg., 193, 1997-2018, (2004) · Zbl 1067.76570 [33] Pontaza, J.P.; Reddy, J.N., Spectral/hp least-squares finite element formulation for the Navier-Stokes equations, J. comp. phys., 190, 523-549, (2003) · Zbl 1077.76054 [34] Russo, A., Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 132, 335-343, (1996) · Zbl 0887.76038 [35] Tezduyar, T.E.; Behr, M.; Liou, J., A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I. the concept and the preliminary numerical tests, Comput. methods appl. mech. engrg., 94, 339-351, (1992) · Zbl 0745.76044 [36] Tezduyar, T.E.; Behr, M.; Mittal, S.; Liou, J., A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. II. computation of free-surface flows, two-liquid flows, and flows with drifting cylinders, Comput. methods appl. mech. engrg., 94, 353-371, (1992) · Zbl 0745.76045 [37] Tezduyar, T.E.; Mittal, S.; Ray, S.E.; Shih, R., Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Comput. methods appl. mech. engrg., 95, 221-242, (1992) · Zbl 0756.76048 [38] Tezduyar, T.E.; Park, Y.J., Discontinuity capturing finite element formulations for nonlinear convection-diffusion-reaction problems, Comput. methods appl. mech. engrg., 59, 307-325, (1986) · Zbl 0593.76096
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