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Hughes, Thomas J. R.

Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. (English) Zbl 0866.76044

Comput. Methods Appl. Mech. Eng. 127, No. 1-4, 387-401 (1995).
Summary: An approach is developed for deriving variational methods capable of representing multiscale phenomena. The ideas are first illustrated on the exterior problem for Helmholtz equation. This leads to the well-known Dirichlet-von-Neumann formulation. Next, a class of subgrid scale models is developed and the relationships to ‘bubble function’ methods and stabilized methods are established. It is shown that both the latter methods are approximate subgrid scale models. The identification for stabilized methods leads to an analytical formula for \(\tau\), the ‘intrinsic time scale’, whose origins have been a mystery heretofore.

Cited in 7 Reviews
Cited in 824 Documents

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76M30 Variational methods applied to problems in fluid mechanics

Keywords:

exterior problem for Helmholtz equation; bubble function methods; intrinsic time scale
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\textit{T. J. R. Hughes}, Comput. Methods Appl. Mech. Eng. 127, No. 1--4, 387--401 (1995; Zbl 0866.76044)
Full Text: DOI

References:

[1] Givoli, D., Numerical Methods for Problems in Infinite Domains (1992), Elsevier: Elsevier Amsterdam · Zbl 0788.76001
[2] Givoli, D.; Keller, J. B., A finite element method for large domains, Comput. Methods Appl. Mech. Engrg., 76, 41-66 (1989) · Zbl 0687.73065
[3] Keller, J. B.; Givoli, D., An exact non-reflecting boundary condition, J. Comput. Phys., 82, 172-192 (1988) · Zbl 0671.65094
[4] Harari, I.; Hughes, T. J.R., Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains, Comput. Methods Appl. Mech. Engrg., 97, 103-124 (1992) · Zbl 0769.76063
[5] Harari, I.; Hughes, T. J.R., Studies of domain-based formulations for computing exterior problems of acoustics, Int. J. Numer. Methods Engrg., 37, 2935-2950 (1994) · Zbl 0818.76040
[6] Brezzi, F.; Bristeau, M. O.; Franca, L. P.; Mallet, M.; Roge, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. Methods Appl. Mech. Engrg., 96, 117-129 (1992) · Zbl 0756.76044
[7] Franca, L. P.; Frey, S. L., Stabilized finite element methods: II. The incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 99, 209-233 (1992) · Zbl 0765.76048
[8] Baiocchi, C.; Brezzi, F.; Franca, L. P., Virtual bubbles and the Galerkin-least-squares method, Comput. Methods Appl. Mech. Engrg., 105, 125-141 (1993) · Zbl 0772.76033
[9] Franca, L. P.; Farhat, C., On the limitations of bubble functions, Comput. Methods Appl. Mech. Engrg., 117, 225-230 (1994) · Zbl 0847.76033
[10] Franca, L. P.; Farhat, C., Anti-stabilizing effects of bubble functions, (Proc. Third World Congress on Computational Mechanics. Proc. Third World Congress on Computational Mechanics, Extended Abstracts, Chiba, Japan, Vol. 2 (August, 1994)), 1452-1453
[11] Franca, L. P.; Farhat, C., Bubble functions prompt unusual stabilized finite element methods, Comput. Methods Appl. Mech. Engrg., 123, 299-308 (1995) · Zbl 1067.76567
[12] Brooks, A.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 199-259 (1982) · Zbl 0497.76041
[13] Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method (1987), Cambridge University Press: Cambridge University Press Cambridge
[14] Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. Comp., 52, 495-508 (1989) · Zbl 0669.76051
[15] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0788.73002
[16] Hughes, T. J.R.; Franca, L. P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59, 85-99 (1986) · Zbl 0622.76077
[17] Hughes, T. J.R.; Franca, L. P., A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg., 65, 85-96 (1987) · Zbl 0635.76067
[18] Franca, L. P.; Hughes, T. J.R.; Loula, A. F.D.; Miranda, I., A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation, Numerische Mathematik, 53, 123-141 (1988) · Zbl 0656.73036
[19] Franca, L. P.; Hughes, T. J.R., Two classes of mixed finite element methods, Comput. Methods Appl. Mech. Engrg., 69, 89-129 (1988) · Zbl 0651.65078
[20] Hughes, T. J.R.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg., 73, 173-189 (1989) · Zbl 0697.76100
[21] Hughes, T. J.R.; Brezzi, F., On drilling degrees-of-freedom, Comput. Methods Appl. Mech. Engrg., 72, 105-121 (1989) · Zbl 0691.73015
[22] Barbosa, H. J.C.; Hughes, T. J.R., Boundary Lagrange multipliers in finite element methods: Error analysis in natural norms, Numerische Mathematik, 62, 1-15 (1992) · Zbl 0765.65102
[23] Franca, L. P.; Hughes, T. J.R., Convergence analysis of Galerkin/least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 105, 285-298 (1993) · Zbl 0771.76037
[24] Franca, L. P.; Hughes, T. J.R.; Stenberg, R., Stabilized finite element methods for the Stokes problem, (Nicolaides, R. A.; Gunzberger, M. D., Incompressible Fluid Dynamics—Trends and Advances (1993), Cambridge University Press: Cambridge University Press Cambridge), 87-107 · Zbl 1189.76339
[25] Hughes, T. J.R.; Hauke, G.; Jansen, K., Stabilized finite element methods in fluids: Inspirations, origins, status and recent developments, (Hughes, T. J.R.; Oñate, E.; Zienkiewicz, O. C., Recent Developments in Finite Element Analysis. A Book Dedicated to Robert L. Taylor (1994), International Center for Numerical Methods in Engineering: International Center for Numerical Methods in Engineering Barcelona, Spain), 272-292 · Zbl 1122.76346
[26] Franca, L. P.; Frey, S. L.; Hughes, T. J.R., Stabilized finite element methods: I. Application to the advective-diffusive model, Comput. Methods Appl. Mech. Engrg., 95, 253-276 (1992) · Zbl 0759.76040
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