Bubble-based stabilization for the Helmholtz equation

International Journal for Numerical Methods in Engineering

Volumn 70, Issue 10, 2007, Pages 1241-1260

  Harari, Isaac ^a   Gosteev, Kirill ^a  

^a TEL AVIV UNIVERSITY   (Israel)

Author keywords

Acoustic; Bubbles; Finite elements; Stabilized methods

Indexed keywords

ACOUSTIC WAVE SCATTERING; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; HELMHOLTZ EQUATION;

DISPERSION ANALYSIS; TIME-HARMONIC ACOUSTICS;

ACOUSTICS;

ACOUSTIC WAVE SCATTERING; ACOUSTICS; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; HELMHOLTZ EQUATION;

EID: 34249903890     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1930     Document Type: Article

References (28)

1. Babuška I, Ihlenburg F, Paik ET, Sauter SA. A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Computer Methods in Applied Mechanics and Engineering 1995; 128(3-4):325-359.
2. Arnold DN, Brezzi F, Fortin M. A stable finite element for the Stokes equations. Calcolo 1984; 21(4):337-344.
3. Brezzi F, Bristeau MO, Franca LP, Mallet M, Rogé G. A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Computer Methods in Applied Mechanics and Engineering 1992; 96(1):117-129.
4. Franca LP, Farhat C. Bubble functions prompt unusual stabilized finite element methods. Computer Methods in Applied Mechanics and Engineering 1995; 123(1-4):299-308.
5. Brezzi F, Franca LP, Russo A. Further considerations on residual-free bubbles for advective-diffusive equations. Computer Methods in Applied Mechanics and Engineering 1998; 166(1-2):25-33.
6. Franca LP, Farhat C, Macedo AP, Lesoinne M. Residual-free bubbles for the Helmholtz equation. International Journal for Numerical Methods in Engineering 1997; 40(21):4003-4009.
7. Franca LP, Nesliturk A, Stynes M. On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method. Computer Methods in Applied Mechanics and Engineering 1998; 166(1-2):35-49.
8. Hughes TJR. Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Computer Methods in Applied Mechanics and Engineering 1995; 127(1-4):387-401.
9. Hughes TJR, Feijóo GR, Mazzei L, Quincy J-B. The variational multiscale method---a paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering 1998; 166(1-2):3-24.
10. Brezzi F, Franca LP, Hughes TJR, Russo A. b = f g. Computer Methods in Applied Mechanics and Engineering 1997; 145(3-4):329-339.
11. 1-optimal finite element methods. Computer Methods in Applied Mechanics and Engineering 2001; 190(43-44):5679-5690.
12. Givoli D. Non-local and semi-local optimal weighting functions for symmetric problems involving a small parameter. International Journal for Numerical Methods in Engineering 1988; 26(6):1281-1298.
13. Brezzi F, Russo A. Choosing bubbles for advection-diffusion problems. Mathematical Models and Methods in Applied Sciences 1994; 4(4):571-587.
14. Nesliturk A, Harari I. The nearly-optimal Petrov-Galerkin method for convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering 2003; 192(22-24):2501-2519.
15. Franca LP, Madureira AL, Valentin F. Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions. Computer Methods in Applied Mechanics and Engineering 2005; 1940(27-29): 3006-3021.
16. Hou TY, Wu X-H. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of Computational Physics 1997; 134(1):169-189.
17. Hughes TJR, Sangalli G. Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and stabilized methods. ICES Report 05-46, Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas, 2005; SIAM Journal on Numerical Analysis, accepted.
18. Franca LP, Frey SL, Hughes TJR. Stabilized finite element methods, I. Application to the advective-diffusive model. Computer Methods in Applied Mechanics and Engineering 1992; 95(2):253-276.
19. Franca LP, Valentin F. On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation. Computer Methods in Applied Mechanics and Engineering 2000; 189(13-14):1785-1800.
20. Oberai AA, Pinsky PM. A residual-based finite element method for the Helmholtz equation. International Journal for Numerical Methods in Engineering 2000; 49(3):399-419.
21. Farhat C, Harari I, Franca LP. The discontinuous enrichment method. Computer Methods in Applied Mechanics and Engineering 2001; 190(48):6455-6479.
22. Farhat C, Harari I, Hetmaniuk U. The discontinuous enrichment method for multiscale analysis. Computer Methods in Applied Mechanics and Engineering 2003; 192(28-30):3195-3209.
23. Gravemeier V, Wall WA, Ramm E. A three-level finite element method for the instationary incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 2004; 193(15-16): 1323-1366.
24. Deraemaeker A, Babuška I, Bouillard P. Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. International Journal for Numerical Methods in Engineering 1999; 46(4):471-499.
25. Farhat C. Private communication. 1996.
26. Gosteev K. Improved finite element methods for acoustics. Master's thesis, Tel Aviv University, Tel Aviv, Israel, 2005.
27. Thompson LL, Pinsky PM. A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation. International Journal for Numerical Methods in Engineering 1995; 38(3):371-397.
28. Harari I, Hughes TJR. Galerkin/least-squares finite element methods for the reduced wave equation with nonreflecting boundary conditions in unbounded domains. Computer Methods in Applied Mechanics and Engineering 1992; 98(3):411-454.