The variational multiscale method (VMS) is a technique used for deriving models and numerical methods for multiscale phenomena.[1] The VMS framework has been mainly applied to design stabilized finite element methods in which stability of the standard Galerkin method is not ensured both in terms of singular perturbation and of compatibility conditions with the finite element spaces.[2]
Stabilized methods are getting increasing attention in computational fluid dynamics because they are designed to solve drawbacks typical of the standard Galerkin method: advection-dominated flows problems and problems in which an arbitrary combination of interpolation functions may yield to unstable discretized formulations.[3][4] The milestone of stabilized methods for this class of problems can be considered the Streamline Upwind Petrov-Galerkin method (SUPG), designed during 80s for convection dominated-flows for the incompressible Navier–Stokes equations by Brooks and Hughes.[5][6] Variational Multiscale Method (VMS) was introduced by Hughes in 1995.[7] Broadly speaking, VMS is a technique used to get mathematical models and numerical methods which are able to catch multiscale phenomena;[1] in fact, it is usually adopted for problems with huge scale ranges, which are separated into a number of scale groups.[8] The main idea of the method is to design a sum decomposition of the solution as , where is denoted as coarse-scale solution and it is solved numerically, whereas represents the fine scale solution and is determined analytically eliminating it from the problem of the coarse scale equation.[1]
^ abcHughes, T.J.R.; Scovazzi, G.; Franca, L.P. (2004). "Chapter 2: Multiscale and Stabilized Methods". In Stein, Erwin; de Borst, René; Hughes, Thomas J.R. (eds.). Encyclopedia of Computational Mechanics. John Wiley & Sons. pp. 5–59. ISBN 0-470-84699-2.
^Codina, R.; Badia, S.; Baiges, J.; Principe, J. (2017). "Chapter 2: Variational Multiscale Methods in Computational Fluid Dynamics". In Stein, Erwin; de Borst, René; Hughes, Thomas J.R. (eds.). Encyclopedia of Computational Mechanics Second Edition. John Wiley & Sons. pp. 1–28. ISBN 9781119003793.
^Masud, Arif (April 2004). "Preface". Computer Methods in Applied Mechanics and Engineering. 193 (15–16): iii–iv. doi:10.1016/j.cma.2004.01.003.
^Cite error: The named reference Quarteroni was invoked but never defined (see the help page).
^Brooks, Alexander N.; Hughes, Thomas J.R. (September 1982). "Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations". Computer Methods in Applied Mechanics and Engineering. 32 (1–3): 199–259. Bibcode:1982CMAME..32..199B. doi:10.1016/0045-7825(82)90071-8.
^Masud, Arif; Calderer, Ramon (3 February 2009). "A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations". Computational Mechanics. 44 (2): 145–160. Bibcode:2009CompM..44..145M. doi:10.1007/s00466-008-0362-3. S2CID 7036642.
^Cite error: The named reference Hughes_2005 was invoked but never defined (see the help page).
^Rasthofer, Ursula; Gravemeier, Volker (27 February 2017). "Recent Developments in Variational Multiscale Methods for Large-Eddy Simulation of Turbulent Flow". Archives of Computational Methods in Engineering. 25 (3): 647–690. doi:10.1007/s11831-017-9209-4. hdl:20.500.11850/129122. S2CID 29169067.
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