×

zbMATH — the first resource for mathematics

A two-level variational multiscale method for convection-dominated convection-diffusion equations. (English) Zbl 1124.76028
Summary: This paper studies the error in, the efficient implementation of and time stepping methods for a variational multiscale method (VMS) for solving convection-dominated problems. The VMS studied uses a fine mesh \(C^{0}\) finite element space \(X^{h}\) to approximate the concentration and a coarse mesh discontinuous vector finite element space \(L^{H}\) for the large scales of the flux in the two scale discretization. Our tests show that these choices lead to an efficient VMS whose complexity is further reduced if a (locally) \(L^{2}\)-orthogonal basis for \(L^{H}\) is used. A fully implicit and a semi-implicit treatment of the terms which link effects across scales are tested and compared. The semi-implicit VMS was much more efficient. The observed global accuracy of the most straightforward VMS implementation was much better than the artificial diffusion stabilization and comparable to a streamline-diffusion finite element method in our tests.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76M30 Variational methods applied to problems in fluid mechanics
76R99 Diffusion and convection
Software:
MooNMD
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anitescu, M.; Layton, W.; Pahlevani, F., Implicit for local effects and explicit for nonlocal effects is unconditionally stable, Etna, 18, 174-187, (2004) · Zbl 1085.76046
[2] Chen, G.Q.; Du, Q.; Tadmor, E., Spectral viscosity approximations for multi-dimensional scalar conservation laws, Math. comput., 61, 629-643, (1993) · Zbl 0799.35148
[3] Codina, R., Comparison of some finite element methods for solving the diffusion-convection-reaction equation, Comput. methods appl. mech. engrg., 156, 185-210, (1998) · Zbl 0959.76040
[4] Codina, R., Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods, Comput. methods appl. mech. engrg., 13, 505-512, (2000)
[5] Fortin, M.; Guenette, R.; Pierre, R., Numerical analysis of the EVSS method, Comput. methods appl. mech. engrg., 143, 79-95, (1997) · Zbl 0896.76040
[6] 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
[7] V. Gravemeier, The variational multiscale method for laminar and turbulent incompressible flow, Ph.D. thesis, Institute of Structural Mechanics, University of Stuttgart, 2003. · Zbl 1177.76341
[8] Gravemeier, V.; Wall, W.A.; Ramm, E., A three-level finite element method for the instationary incompressible Navier-Stokes equation, Comput. methods appl. mech. engrg., 193, 1323-1366, (2004) · Zbl 1085.76038
[9] Guermond, J.L., Stabilization of Galerkin approximations of transport equations by subgrid modelling, Math. modell. numer. anal., 1293-1316, (1999) · Zbl 0946.65112
[10] N. Heitmann, Subgrid stabilization of evolutionary diffusive transport problems, Ph.D. thesis, University of Pittsburgh, 2004.
[11] Hughes, T.J.; Mazzei, L.; Jensen, K.E., Large eddy simulation and the variational multiscale method, Comput. vis. sci., 3, 47-59, (2000) · Zbl 0998.76040
[12] Hughes, T.J.; Oberai, A.; Mazzei, L., Large eddy simulation of turbulent channel flows by the variational multiscale method, Phys. fluids, 13, 1784-1799, (2001) · Zbl 1184.76237
[13] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origin of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044
[14] Hughes, T.J.R.; Brooks, A.N., A multidimensional upwind scheme with no crosswind diffusion, (), 19-35 · Zbl 0423.76067
[15] Hughes, T.J.R.; Mazzei, L.; Oberai, A.A.; Wray, A.A., The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence, Phys. fluids, 13, 505-512, (2001) · Zbl 1184.76236
[16] John, V., Large eddy simulation of turbulent incompressible flows, ()
[17] V. John, S. Kaya, Finite element error analysis of a variational multiscale method for the Navier-Stokes equations, Adv. Comput. Math., in press. · Zbl 1126.76030
[18] John, V.; Kaya, S., A finite element variational multiscale method for the Navier-Stokes equations, SIAM J. sci. comput., 26, 1485-1503, (2005) · Zbl 1073.76054
[19] John, V.; Matthies, G., Moonmd—a program package based on mapped finite element methods, Comput. vis. sci., 6, 163-170, (2004) · Zbl 1061.65124
[20] Johnston, H.; Liu, J.-L., Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, J. comput. phys., 199, 221-259, (2004) · Zbl 1127.76343
[21] S. Kaya, Numerical analysis of subgrid-scale eddy viscosity for the Navier-Stokes equations, Technical Report, University of Pittsburgh, 2002.
[22] S. Kaya, W. Layton, Subgrid-scale eddy viscosity methods are variational multiscale methods, Technical Report, University of Pittsburgh, 2002.
[23] Kaya, S.; Riviere, B., A discontinuous subgrid eddy viscosity method for the time dependent Navier-Stokes equations, SIAM J. numer. anal., 43, 1572-1595, (2005) · Zbl 1096.76026
[24] S. Kaya, B. Riviere, A two-grid stabilization method for solving the steady-state Navier-Stokes equations, Numer. Methods Part. Differen. Equat., in press. · Zbl 1089.76034
[25] W. Layton, Variational multiscale methods and subgrid scale eddy viscosity, in: H. Deconinck (Ed.), 32nd, 2002-2006, Computational Fluid Dynamics Multiscale Methods, Von Karman Institute for Fluid Dynamics, Rhode-Saint-Genèse, Belgium, 2002.
[26] Layton, W.J., A connection between subgrid scale eddy viscosity and mixed methods, Appl. math. comput., 133, 147-157, (2002) · Zbl 1024.76026
[27] Lonsdale, R., An algebraic multigrid solver for the Navier-Stokes equations on unstructured meshes, Int. J. numer. methods heat fluid flow, 3, 3-14, (1993)
[28] Maday, Y.; Tadmor, E., Analysis of the spectral vanishing method for periodic conservation laws, SIAM J. numer. anal., 26, 854-870, (1989) · Zbl 0678.65066
[29] F. Pahlevani, Sensitivity analysis of eddy viscosity models, Ph.D. thesis, University of Pittsburgh, 2004. · Zbl 1106.76049
[30] Roos, H.-G.; Stynes, M.; Tobiska, L., Numerical methods for singularly perturbed differential equations, (1996), Springer
[31] Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. sci. comput., 14, 2, 461-469, (1993) · Zbl 0780.65022
[32] Smagorinsky, J.S., General circulation experiments with the primitive equations, Mon. weather rev., 91, 99-164, (1963)
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.