Geodesic
Anisotropic mesh adaption: application to computational fluid dynamics
Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 1, pp. 145-165.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

In this communication we focus on goal-oriented anisotropic adaption techniques. Starting point has been the derivation of suitable anisotropic interpolation error estimates for piecewise linear finite elements, on triangular grids in $2D$. Then we have merged these interpolation estimates with the dual-based a posteriori error analysis proposed by R. Rannacher and R. Becker. As examples of this general anisotropic a posteriori analysis, elliptic, advection-diffusion-reaction and the Stokes problems are analyzed. Finally, numerical test cases are provided to assess the soundness of the proposed approach.
In questa comunicazione vengono presentate tecniche di adattazione di griglia goal-oriented di tipo anisotropo. Punto di partenza è stata la derivazione di opportune stime di tipo anisotropo per l'errore d'interpolazione, per elementi finiti lineari a pezzi, su griglie triangolari in $2D$. Si sono quindi utilizzate tali stime d'interpolazione per generalizzare al caso anisotropo l'analisi a posteriori proposta da R. Rannacher e da R. Becker, basata su un approccio di tipo duale. In questo lavoro tale analisi a posteriori viene particolarizzata al caso di problemi ellittici, di trasporto-diffusione-reazione e al problema di Stokes. Vengono da ultimo forniti alcuni risultati numerici al fine di validare l’affidabilità dell’approccio proposto 
@article{BUMI_2005_8_8B_1_a7,
     author = {Perotto, Simona},
     title = {Anisotropic mesh adaption: application to computational fluid dynamics},
     journal = {Bollettino della Unione matematica italiana},
     pages = {145--165},
     publisher = {mathdoc},
     volume = {Ser. 8, 8B},
     number = {1},
     year = {2005},
     zbl = {1150.65028},
     mrnumber = {1885308},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_1_a7/}
}
TY  - JOUR
AU  - Perotto, Simona
TI  - Anisotropic mesh adaption: application to computational fluid dynamics
JO  - Bollettino della Unione matematica italiana
PY  - 2005
SP  - 145
EP  - 165
VL  - 8B
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_1_a7/
LA  - en
ID  - BUMI_2005_8_8B_1_a7
ER  - 
%0 Journal Article
%A Perotto, Simona
%T Anisotropic mesh adaption: application to computational fluid dynamics
%J Bollettino della Unione matematica italiana
%D 2005
%P 145-165
%V 8B
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_1_a7/
%G en
%F BUMI_2005_8_8B_1_a7
Perotto, Simona. Anisotropic mesh adaption: application to computational fluid dynamics. Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 1, pp. 145-165. http://geodesic.mathdoc.fr/item/BUMI_2005_8_8B_1_a7/

[1] M. Ainsworth - J. T. Oden, A posteriori error estimation in finite element analysis, John Wiley & Sons, Inc., New-York, 2000. | MR | Zbl

[2] R. C. Almeida - R. A. Feijóo - A. C. Galeão - C. Padra - R. S. Silva, Adaptive finite element computational fluid dynamics using an anisotropic error estimator, Comput. Methods Appl. Mech. Engrg., 182 (2000), 379-400. | MR | Zbl

[3] T. Apel, Anisotropic finite elements: local estimates and applications, Book Series: Advances in Numerical Mathematics, Teubner, Stuttgart, 1999. | MR | Zbl

[4] T. Apel - G. Lube, Anisotropic mesh refinement in stabilized Galerkin methods, Numer. Math., 74 (1996), 261-282. | MR | Zbl

[5] I. Babuška - W. Rheinboldt, A posteriori error estimates for the finite element method, Int. J. Numer. Methods Eng., 12 (1978), 1597-1615. | Zbl

[6] R. E. Bank - A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44, no. 170 (1985), 283-301. | MR | Zbl

[7] R. Becker - R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica, 10 (2001), 1-102. | MR | Zbl

[8] F. Brezzi - A. Russo, Choosing bubbles for advection-diffusion problems, Math. Models Methods Appl. Sci., 4 (1994), 571-587. | MR | Zbl

[9] A. N. Brooks - T. J. R. Hughes, Streamline upwind / Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), 199-259. | MR | Zbl

[10] Ph. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Company, Amsterdam, 1978. | MR | Zbl

[11] Ph. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numér., 2 (1975), 77-84. | fulltext mini-dml | MR | Zbl

[12] D. L. Darmofal - D. A. Venditti, Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows, J. Comput. Phys., 187 (2003) 22-46. | Zbl

[13] E. F. D’Azevedo - R. B. Simpson, On optimal triangular meshes for minimizing the gradient error, Numer. Math., 59 (1991), 321-348. | MR | Zbl

[14] J. Douglas - J. Wang, An absolutely stabilized finite element method for the Stokes problem, Math. Comp., 52 (1989), 495-508. | MR | Zbl

[15] K. Eriksson - C. Johnson, Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems, Math. Comp., 60 (1993), 167-188. | MR | Zbl

[16] K. Eriksson - D. Estep - P. Hansbo - C. Johnson, Introduction to adaptive methods for differential equations, Acta Numerica, (1995), 105-158. | MR | Zbl

[17] L. Formaggia - S. Perotto, New anisotropic a priori error estimates, Numer. Math., 89 (2001), 641-667. | MR | Zbl

[18] L. Formaggia - S. Perotto, Anisotropic error estimates for elliptic problems, Numer. Math., 94 (2003), 67-92. | MR | Zbl

[19] L. Formaggia - S. Micheletti - S. Perotto, Anisotropic mesh adaptation in Computational Fluid Dynamics: application to the advection-diffusion-reaction and the Stokes problems, to appear in Appl. Num. Math. (2004). | MR | Zbl

[20] L. Formaggia - S. Perotto - P. Zunino, An anisotropic a-posteriori error estimate for a convection-diffusion problem, Comput. Visual. Sci., 4 (2001), 99-104. | MR | Zbl

[21] L. P. Franca - T. J. R. Hughes, Convergence analyses of Galerkin least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 105 (1993), 285-298. | MR | Zbl

[22] L. Franca - R. Stenberg, Error analysis of some GLS methods for elasticity equations, SIAM J. Numer. Anal., 28 (1991), 1680-1697. | MR | Zbl

[23] F. Courty - D. Leservoisier - P. L. George - A. Dervieux, Continuous metrics and mesh optimization, submitted for the publication in Appl. Numer. Math., (2003).

[24] P. L. George - H. Borouchaki, Delaunay triangulation and meshing-application to finite element, Editions Hermes, Paris, 1998. | MR | Zbl

[25] M. B. Giles - E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numerica, 11 (2002), 145-236. | MR | Zbl

[26] W. G. Habashi - M. Fortin - J. Dompierre - M. G. Vallet - Y. Bourgault, Anisotropic mesh adaptation: a step towards a mesh-independent and user-independent CFD, in Barriers and Challenges in Computational Fluid Dynamics, Kluwer Acad. Publ., 1998, 99-117. | MR | Zbl

[27] F. Hecht, BAMG: bidimensional anisotropic mesh generator, (1998). http://www-rocq.inria.fr/gamma/cdrom/www/bamg/eng.htm

[28] T. J. R. Hughes - L. Franca - M. Balestra, 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 equalorder interpolations, Comput. Methods Appl. Mech. Engrg., 59 (1986), 85-99. | MR | Zbl

[29] T. J. R. Hughes - L. P. Franca - G. M. Hulbert, 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 (1989), 173-189. | MR | Zbl

[30] G. Kunert, A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes, Ph.D. thesis, Fakultät für Mathematik der Technischen Universität Chemnitz, Chemnitz, 1999. | Zbl

[31] J. L. Lions - E. Magenes, Non-homogeneous boundary value problem and application, Volume I. Springer-Verlag, Berlin, 1972. | MR | Zbl

[32] S. Micheletti - S. Perotto - M. Picasso, Stabilized finite elements on anisotropic meshes: a priori error estimates for the advection-diffusion and Stokes problems, SIAM J. Numer. Anal., 41, no. 3 (2003), 1131-1162. | MR | Zbl

[33] S. Mittal, On the performance of high aspect ratio elements for incompressible flows, Comput. Methods Appl. Mech. Engrg., 188 (2000), 269-287. | Zbl

[34] J. T. Oden - S. Prudhomme, Goal-oriented error estimation and adaptivity for the finite element method, Computers Math. Applic., 41, no. 5-6 (2001), 735-756. | MR | Zbl

[35] M. Picasso, An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: application to elliptic and parabolic problems, SIAM J. Sci. Comput., 24, no. 4 (2003), 1328-1355. | MR | Zbl

[36] L. R. Scott - S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions, Math. Comp., 54 (1990), 483-493. | MR | Zbl

[37] K. G. Siebert, An a posteriori error estimator for anisotropic refinement, Numer. Math., 73 (1996), 373-398. | MR | Zbl

[38] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, B. G. Teubner, Stuttgart, 1996. | Zbl

[39] O. C. Zienkiewicz - J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Methods Eng., 24 (1987), 337-357. | MR | Zbl

[40] O. C. Zienkiewicz - J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates, Part 1: the recovery technique, Int. J. Numer. Methods Eng., 33 (1992), 1331-1364. | MR | Zbl

[41] O. C. Zienkiewicz - J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates, Part 2: error estimates and adaptivity, Int. J. Numer. Methods Eng., 33 (1992), 1365-1382. | MR | Zbl