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A connection between subgrid scale eddy viscosity and mixed methods. (English) Zbl 1024.76026
Summary: We consider a new mixed method (related to the EVSS method in computational viscoelasticity) for the convection-dominated, convection-diffusion equation in which stabilization is added then removed through the extra ‘mixed’ variables. This consistent stabilization is equivalent to an artificial viscosity operator acting only on the fluctuations in \(\nabla u^h\). By suitable choice of the mixed spaces, a method of Guermond [J.-L. Guermond, M2AN, Math. Model. Numer. Anal. 33, 1293-1316 (1999; Zbl 0946.65112)] is recovered exactly. We show that for a different, natural choice a new method results with global error estimates similar to both Guermond’s method and the streamline diffusion/SUPG method.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76R50 Diffusion
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[1] Babuska, I.; Aziz, A.K., Survey lectures on the mathematical foundation of the finite element method, () · Zbl 0268.65052
[2] Bernardi, C.; Canuto, C.; Mayday, Generalized inf – sup conditions for Chebyshev spectral approximations of the Stokes problem, SIAM J. numer. anal., 25, 1237-1265, (1988)
[3] Brezzi, F.; Fortin, M., Mixed and hybrid finite elements methods, (1991), Springer Berlin
[4] 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, Comput. meth. appl. mech. eng., 32, 199-259, (1982) · Zbl 0497.76041
[5] Ervin, V.; Layton, W.; Maubach, J., Adaptive defect correction methods for viscous incompressible flow problems, (), 1165-1185 · Zbl 1049.76038
[6] Fortin, M.; Guenette, R.; Pierre, R., Numerical analysis of the EVSS method, Comput. meth. appl. mech. eng., 143, 79-95, (1997) · Zbl 0896.76040
[7] L. Franca, A. Nesliturk, On a two-level finite element method for the incompressible Navier-Stokes equations, preprint (2000) · Zbl 1002.76066
[8] Guermond, J.-L., Stabilization of Galerkin approximations of transport equations by subgrid modeling, M2an, 33, 1293-1316, (1999) · Zbl 0946.65112
[9] Guermond, J.-L., Stabilisation par viscosité de sous-maille pour l’approximation de Galerkin des opérateurs linéaires montones, C.r.a.s., 328, 617-622, (1999) · Zbl 0933.65058
[10] Gunzburger, M., Finite element methods for viscous incompressible flows, (1989), Academic Press San Diego, CA · Zbl 0697.76031
[11] Hughes, T.J.; Mazzei, L.; Jansen, K.E., Large eddy simulation and the variational multiscale method, Comput. visual sci., 3, 47-59, (2000) · Zbl 0998.76040
[12] T. Iliescu, Genuinely nonlinear models for convection dominated problems, Preprint ANL/MCS-P857-1100, Argonne National Lab., 2000 · Zbl 1329.76173
[13] Iliescu, T.; Layton, W., Approximating the larger eddies in fluid motion III: the Boussinesq model for turbulent fluctuations, (), 245-261 · Zbl 1078.76553
[14] Johnson, C., Numerical solution of partial differential equations by the finite element methods for stationary convection – diffusion problems, (1987), Cambridge University Press Cambridge
[15] Layton, W., A nonlinear, subgridscale model for incompressible viscous flow problems, SIAM J. sci. comput., 17, 347-357, (1996) · Zbl 0844.76054
[16] Nicolaides, R.A., Existence uniqueness and approximation for generalized saddle point problems, SIAM J. numer. anal., 19, 349-357, (1982) · Zbl 0485.65049
[17] H.-G. Roos, T. Linss, Sufficient conditions for uniform convergence on layer-adapted grids, pre-print, 1998 · Zbl 0931.65085
[18] Roos, H.-G.; Stynes, M.; Tobiska, L., Numerical methods for singularly perturbed differential equations, (1996), Springer Berlin
[19] Smagorinski, J., General circulation experiments with the primitive equations, Mon. weather rev., 91, 216-241, (1963)
[20] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
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