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Subgrid Phenomena and Numerical Schemes

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Frontiers in Numerical Analysis

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Abstract

In recent times, several attempts have been mad e to recover some information from the subgrid scales and transfer them to the computational scales. Many stabilizing techniques can also be considered as part of this effort. We discuss here a framework in which some of these attempts can be set and analyzed.

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References

  1. T. Arbogast, “Numerical subgrid upscaling of two-phase flow in porous media,” in “Multiphase flows and transport in porous media: State of the art”, (Z. Chen, R.E. Ewing, and Z.-C. Shi eds.), Lecture Notes in Physics, Springer, Berlin, 2000.

    Google Scholar 

  2. T. Arbogast, S.E. Minkoff, and P.T. Keenan, “An operator-based approach to upscaling the pressure equation,” in: Computational Methods in Water Resources XII, v.1, V.N. Burganos et als., eds., Computational Mechanics Publications, Southampton, U.K., 1998.

    Google Scholar 

  3. F. Brezzi, M. Fortin, “Mixed and Hybrid Finite Element Methods,” Springer Verlag, New York, Springer Series in Computational Mathematics 15, 1991.

    Google Scholar 

  4. F. Brezzi, L.P. Franca, T.J.R. Hughes, and A. Russo, “b = ∫ g,” Comput. Methods Appl. Mech. Engrg. 145, 329–339 (1997). Methods Appl. Mech. Engrg. 166, 25–33 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Brezzi, P. Houston, L.D. Marini, and E. Süli, “Modeling subgrid viscosity for advection-diffusion problems,” Comput. Methods Appl. Mech. Engrg. 190, 1601–1610 (2000).

    Article  MATH  Google Scholar 

  6. F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Russo, and E. Süli, “A priori error analysis of a finite element method with residual-free bubbles for advection-dominated equations,” SIAM J. Num. Anal. 36, 1933–1948 (1999)

    Article  MATH  Google Scholar 

  7. J. Bergh, J. Löfström “Interpolation Spaces” Springer Verlag, Berlin, 1976.

    Book  MATH  Google Scholar 

  8. F. Brezzi, D. Marini, and A. Russo, “Applications of pseudo residual-free bubbles to the stabilization of convection-diffusion problems,” Comput. Methods Appl. Mech. Engrg. 166, 51–63 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Brezzi, D. Marini, and E. Süli, “Residual-free bubbles for advection-diffusion problems: the general error analysis,” Numer. Math. 85, 31–47 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Brezzi, A. Russo, “Choosing bubbles for advection-diffusion problems,” Math. Mod. and Meth. in Appl. Sci. 4, 571–587 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  11. A.N. Brooks, 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).

    Article  MathSciNet  MATH  Google Scholar 

  12. Ph.G. Ciarlet, “The finite element method for elliptic problems,” North-Holland, 1978.

    Google Scholar 

  13. C. Farhat, A. Macedo, and M. Lesoinne, “A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems,” Numer. Math. 85, 283–308 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. L.P. Franca, C. Farhat, A.P. Macedo and M. Lesoinne, “Residual-Free Bubbles for the Helmholtz Equation,” Int. J. Num. Meth. in Eng. 40, 4003–4009 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  15. L.P. Franca, S.L. Frey and T.J.R. Hughes, “Stabilized finite element methods: I. Applications to advective-diffusive model,” Comput. Methods Appl, Mech. Engrg. 95, 253–276 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  16. L.P. Franca, A.P. Macedo, “A Two-Level Finite Element Method and its Application to the Helmholtz Equation,” Int. J. Num. Meth. in Eng. 43, 23–32 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  17. L.P. Franca, A. Nesliturk and M. Stynes, “On the Stability of Residual-Free Bubbles for Convection-Diffusion Problems and Their Approximation by a Two-Level Finite Element Method,” Comput. Met hods Appl. Mech. Engrg. 166, 35–49 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  18. L.P. Franca, A. Russo, “Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles.” Appl. Math. Lett. 9, 83–88 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  19. D.F. Griffiths, A.R. Mitchell,“ Spurious behavior and nonlinear instability in discretised partial differential equations,” In: The dyn ami es of numerics and the numerics of dynamics. Inst. Math. Appl. Conf. Ser., New Ser. 34, 215–242 (1992).

    MathSciNet  Google Scholar 

  20. J.L. Guermond, “Stabilization of Galerkin approximations of transport equations by subgrid modeling,” Math. Mod. Num. Anal. 33 (6), 1293–1316 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Hansbo, C. Johnson, “Streamline diffusion finite element methods for fluid flow,” von Karman Institute Lectures, 1995.

    Google Scholar 

  22. T.Y. Hou, X.H. Wu, “A multiscale finite element method for elliptic problems in composite materials and porous media,” J. Comput. Phys. 134, 169–189 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  23. T.Y. Hou, X.H. Wu, and Z. Cai, “Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients,” Math. of Comp. 68, 913–943 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  24. 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).

    Article  MathSciNet  MATH  Google Scholar 

  25. A.R. Mitchell, D.F. Griffiths, “Generalised Galerkin methods for second order equations with significant first derivativeterms,” In: Proc. bienn. Conf., Dundee 1977, Lect. Notes Math 630, 90–104 (1978).

    MathSciNet  Google Scholar 

  26. H.-G. Roos, M. Stynes, and L. Tobiska, “Numerical methods for singularly perturbed differential equations: convection diffusion and flow problems,” Springer Verlag, 1996.

    Google Scholar 

  27. A. Russo, “A posteriori error estimators via bubble functions,” Math. Models Methods Appl. Sci. 6, 353–360 (1997).

    Google Scholar 

  28. G. Sangalli, “Global and local error analysis for t he Residual Free Bubble method applied to advection-dominated problems,” Submitted to Numer. Math.

    Google Scholar 

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Brezzi, F., Marini, D. (2003). Subgrid Phenomena and Numerical Schemes. In: Blowey, J.F., Craig, A.W., Shardlow, T. (eds) Frontiers in Numerical Analysis. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55692-0_1

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  • DOI: https://doi.org/10.1007/978-3-642-55692-0_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44319-3

  • Online ISBN: 978-3-642-55692-0

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