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Stability of the SUPG finite element method for transient advection-diffusion problems. (English) Zbl 1067.76563
Summary: Implicit time integration coupled with SUPG discretization in space leads to additional terms that provide consistency and improve the phase accuracy for convection dominated flows. Recently, it has been suggested that for small Courant numbers these terms may dominate the streamline diffusion term, ostensibly causing destabilization of the SUPG method. While consistent with a straightforward finite element stability analysis, this contention is not supported by computational experiments and contradicts earlier Von-Neumann stability analyses of the semidiscrete SUPG equations. This prompts us to re-examine finite element stability of the fully discrete SUPG equations. A careful analysis of the additional terms reveals that, regardless of the time step size, they are always dominated by the consistent mass matrix. Consequently, SUPG cannot be destabilized for small Courant numbers. Numerical results that illustrate our conclusions are reported.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76R99 Diffusion and convection
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] Bradford, S.F.; Katopodes, N.D., The antidissipative, non-monotone behavior of petrov – galerkin upwinding, Int. J. numer. meth. fluids, 33, 583-608, (2000) · Zbl 0976.76037
[2] Carey, G.; Oden, T., Finite elements. computational aspects, (1984), Prentice-Hall Englewood Cliffs, NJ · Zbl 0558.73064
[3] Ciarlet, P., The finite element method for elliptic problems, (2002), SIAM Philadelphia
[4] Franca, L.P.; Dutra do Carmo, E.G., The Galerkin gradient least-squares method, Comput. methods appl. mech. engrg., 74, 41-54, (1989) · Zbl 0699.65077
[5] Franca, L.P.; Farhat, C., Bubble functions prompt unusual stabilized finite element methods, Comput. methods appl. mech. engrg., 123, 299-308, (1995) · Zbl 1067.76567
[6] Franca, L.P.; Frey, S.; Hughes, T.J.R., Stabilized finite element methods: I. application to the advective – diffusive model, Comput. methods appl. mech. engrg., 95, 253-276, (1992) · Zbl 0759.76040
[7] Franca, L.P.; Valentin, F., On an improved unusual stabilized finite element method for advective – reactive – diffusive equations, Comput. methods appl. mech. engrg., 189, 1785-1800, (2000) · Zbl 0976.76038
[8] I. Harari, Spatial stability of semidiscrete formulations for parabolic problems, in: J. Eberhardsteiner H. Mang, F. Rammerstorfer, (Eds.), Proceedings of the Fifth World Congress on Computational Mechnics, Vienna, Austria, 7-12 July 2002, TU Vienna, Technical University, Vienna
[9] Harari, I., Stability of semidiscrete formulations for parabolic problems at small time steps, Comput. methods appl. mech. engrg., 193, 1491-1516, (2004) · Zbl 1079.76597
[10] Harari, I.; Hughes, T.J.R., What are c and h? inequalities for the analysis and design of finite element methods, Comput. methods appl. mech. engrg., 97, 157-192, (1992) · Zbl 0764.73083
[11] Hughes, T.J.R., Multiscale phenomena: Green’s function, the Dirichlet-to-Neumann map, subgrid scale models, bubbles and the origins of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044
[12] Hughes, T.J.R.; Brooks, A., Streamline upwind/petrov – galerkin formulation 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
[13] Hughes, T.J.R.; Brooks, A., A theoretical framework for petrov – galerkin methods with discontinuous weighting functions: application to the streamline-upwind procedure, (), 47-65
[14] Hughes, T.J.R.; Feijoo, G.R.; Mazzei, L.; Quincy, J.B., The variational multiscale method: a paradigm for computational mechanics, Comput. methods appl. mech. engrg., 166, 3-24, (1998) · Zbl 1017.65525
[15] Hughes, T.J.R.; Franca, L.P.; Hulbert, G., 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, 173-189, (1989) · Zbl 0697.76100
[16] Hughes, T.J.R.; Mallet, M.; Mizukami, A., A new finite element formulation for computational fluid dynamics: II. beyond SUPG, Comput. methods appl. mech. engrg., 54, 341-355, (1986) · Zbl 0622.76074
[17] Hughes, T.J.R.; Stewart, J.R., A space-time formulation for multiscale phenomena, Comput. methods appl. mech. engrg., 74, 217-229, (1995) · Zbl 0869.65061
[18] Ilinica, F.; Hetu, J.-F., Galerkin gradient least-squares formulations for transient conduction heat transfer, Comput. methods. appl. mech. engrg., 191, 3073-3097, (2002) · Zbl 0999.80014
[19] Johnson, C.; Navert, U.; Pitkaranta, J., Finite element methods for linear hyperbolic problems, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[20] Raymond, W.H.; Gardner, A., Selective damping in a Galerkin method for solving wave problems with variable grids, Monthly weather rev., 104, 1583-1590, (1976)
[21] 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, Comput. methods appl. mech. engrg., 87, 35-58, (1991) · Zbl 0760.76051
[22] Shakib, F.; Hughes, T.J.R.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. the compressible Euler and navier – stokes equations, Comput. methods appl. mech. engrg., 89, 141-219, (1991) · Zbl 0838.76040
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