Layton, W. A connection between subgrid scale eddy viscosity and mixed methods. (English) Zbl 1024.76026 Appl. Math. Comput. 133, No. 1, 147-157 (2002). 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. Cited in 3 ReviewsCited in 89 Documents MSC: 76M10 Finite element methods applied to problems in fluid mechanics 76R50 Diffusion Keywords:subgrid scale eddy viscosity; mixed method; convection-diffusion equation; stabilization; artificial viscosity; global error estimates; Guermond’s method; streamline diffusion/SUPG method Citations:Zbl 0946.65112 PDF BibTeX XML Cite \textit{W. Layton}, Appl. Math. Comput. 133, No. 1, 147--157 (2002; Zbl 1024.76026) Full Text: DOI References: [1] Babuska, I.; Aziz, A. K., Survey lectures on the mathematical foundation of the finite element method, (Aziz, A. K., The Mathematical Foundation of the Finite Elements (1972), Academic Press: Academic Press New York) · 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: 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, (SIAM J. Numer. Anal., 37 (2000)), 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 [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: 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 [13] Iliescu, T.; Layton, W., Approximating the larger eddies in fluid motion III: the Boussinesq model for turbulent fluctuations, (Analele Stiintifice ale Universitatii “Al.l.Cuza”. Analele Stiintifice ale Universitatii “Al.l.Cuza”, Series Mathematics, tomul XLIV (1998)), 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 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 [18] Roos, H.-G.; Stynes, M.; Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations (1996), Springer: 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: Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.