Abstract
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.
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References
M. Germano, U. Piomelli, P. Moin, and W.H. Cabot. A dynamic subgrid-scale eddy viscosity model. Phys. Fluid, 3:1760–1765, 1991.
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.
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.
A.A. Oberai and J. Wanderer. Variational formulation of the Germano identity for the Navier-Stokes equations. Journal of Turbulence, 6(7): 2005.
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.
J. Hoffman. Dynamic subgrid modeling for time dependent convection-diffusionreaction equations with fractal solutions. Int. J. Numer. Meth. Fluids, 40:583–592, 2002.
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.
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.
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.
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.
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.
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.
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.
I. Akkerman. Adaptive variational multiscale formulations using the discrete Germano approach. Dissertation, University of Technology Delft, 2009.
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.
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Akkerman, I., Hulshoff, S.J., van der Zee, K.G., de Borst, R. (2010). Variational Germano Approach for Multiscale Formulations. In: de Borst, R., Ramm, E. (eds) Multiscale Methods in Computational Mechanics. Lecture Notes in Applied and Computational Mechanics, vol 55. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9809-2_4
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DOI: https://doi.org/10.1007/978-90-481-9809-2_4
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