Variational Germano Approach for Multiscale Formulations

  • Ido Akkerman
  • Steven J. Hulshoff
  • Kris G. van der Zee
  • René de Borst
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 55)


In this chapter the recently introduced Variational Germano procedure is revisited. The procedure is explained using commutativity diagrams. A general Germano identity for all types of discretizations is derived. This relation is similar to the Variational Germano identity, but is not restricted to variational numerical methods. Based on the general Germano identity an alternative algorithm, in the context of stabilized methods, is proposed. This partitioned algorithm consists of distinct building blocks. Several options for these building blocks are presented and analyzed and their performance is tested using a stabilized finite element formulation for the convectionU? diffusion equation. Non-homogenous boundary conditions are shown to pose a serious problem for the dissipation method. This is not the case for the leastsquares method although here the issue of basis dependence occurs. The latter can be circumvented by minimizing a dual-norm of the weak relation instead of the Euclidean norm of the discrete residual.

Key words

Variational Germano stabilized finite elements variational multiscale 


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  1. 1.
    M. Germano, U. Piomelli, P. Moin, and W.H. Cabot. A dynamic subgrid-scale eddy viscosity model. Phys. Fluid, 3:1760–1765, 1991.zbMATHCrossRefGoogle Scholar
  2. 2.
    A.A. Oberai and J. Wanderer. A dynamic approach for evaluating parameters in a numerical method. Int. J. Numer. Meth. Fluids, 62:50–71, 2005.zbMATHMathSciNetGoogle Scholar
  3. 3.
    A.A. Oberai and J. Wanderer. A dynamic multiscale viscosity method for the spectral approximation of conservation laws. Comput. Methods Appl. Mech. Engrg., 195:1778–1792, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A.A. Oberai and J. Wanderer. Variational formulation of the Germano identity for the Navier-Stokes equations. Journal of Turbulence, 6(7): 2005.zbMATHMathSciNetGoogle Scholar
  5. 5.
    A.A. Oberai and J. Wanderer. Optimal numerical solution of PDEs using the variational Germano identity. Comput. Methods Appl. Mech. Engrg., 197:2948–2962, 2008.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J. Hoffman. Dynamic subgrid modeling for time dependent convection-diffusionreaction equations with fractal solutions. Int. J. Numer. Meth. Fluids, 40:583–592, 2002.zbMATHCrossRefGoogle Scholar
  7. 7.
    E. Onate, J. Garcia, and S. Idelsohn. Computation of the stabilization parameter for the finite element solution of advection-diffusion problems. Int. J. Numer. Meth. Fluids, 25:1385–1407, 1997.zbMATHCrossRefGoogle Scholar
  8. 8.
    T.J.R. Hughes. Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg., 127:387–401, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    T.J.R. Hughes, G.R. Feijoo, L. Mazzei, and J.B. Quincy. The variational multiscale method - A paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg., 166:3–24, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    T.J.R. Hughes and G. Sangalli. Variational multiscale analysis: the fine-scale green’s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal, 45:539–557, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A.N. Brooks and T.J.R. Hughes. Streamline Upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier- Stokes equations. Comput. Methods Appl. Mech. Engrg., 32:199–259, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    L.P. Franca, S.L. Frey, and T.J.R. Hughes. Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg., 95:253–276, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    I. Harari and T.J.R. Hughes. What are C and h?: Inequalities for the analysis and design of finite element method. Comput. Methods Appl. Mech. Engrg., 97:157–192, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    I. Akkerman. Adaptive variational multiscale formulations using the discrete Germano approach. Dissertation, University of Technology Delft, 2009.Google Scholar
  15. 15.
    I. Akkerman, K.G. van der Zee, and S.J. Hulshoff. A Variational Germano approach for Stabilized Finite element methods. Comput. Methods Appl. Mech. Engrg. Accepted.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Ido Akkerman
    • 1
  • Steven J. Hulshoff
    • 2
  • Kris G. van der Zee
    • 3
  • René de Borst
    • 4
  1. 1.Structural EngineeringUniversity of California, San DiegoLa JollaUSA
  2. 2.Aerospace EngineeringDelft University of TechnologyDelftThe Netherlands
  3. 3.Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA
  4. 4.Mechanical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands

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