Journal of Scientific Computing

, Volume 71, Issue 1, pp 386–413 | Cite as

Convergence to Suitable Weak Solutions for a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling

  • Santiago Badia
  • Juan Vicente Gutiérrez-Santacreu


In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup condition.


Navier–Stokes equations Suitable weak solutions Stabilized finite element methods, Subgrid scales 

Mathematics Subject Classification

35Q30 65N30 76N10 



The authors are very grateful to Professor Vivette Girault who provided a proof of a particular case of inequality (8).

This article was funded by Secretaría de Estado de Investigación, Desarrollo e Innovación and European Research Council with Grant Number MTM2015-69875-P and Starting Grant No. 258443 respectively.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Santiago Badia
    • 1
    • 2
  • Juan Vicente Gutiérrez-Santacreu
    • 3
  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Centre Internacional de Mètodes Numèrics en EnginyeriaParc Mediterrani de la TecnologiaCastelldefelsSpain
  3. 3.Dpto. de Matemática Aplicada I, E. T. S. I. InformáticaUniversidad de SevillaSevillaSpain

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