Geodesic
Stabilizations of the Baumann-Oden DG Formulation: the 3D Case
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 629-643.

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

We consider the Baumann-Oden Discontinuous Galerkin formulation in three dimensions in a rather general geometrical setting. Using only piecewise linear approximations (and no jump stabilizations) the method is clearly unstable. We discuss the relations of possible jump stabilizations and bubble stabilizations. 
@article{BUMI_2008_9_1_3_a6,
     author = {Antonietti, Paola F. and Brezzi, Franco and Marini, L.Donatella},
     title = {Stabilizations of the {Baumann-Oden} {DG} {Formulation:} the {3D} {Case}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {629--643},
     publisher = {mathdoc},
     volume = {Ser. 9, 1},
     number = {3},
     year = {2008},
     zbl = {1192.65141},
     mrnumber = {2455336},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a6/}
}
TY  - JOUR
AU  - Antonietti, Paola F.
AU  - Brezzi, Franco
AU  - Marini, L.Donatella
TI  - Stabilizations of the Baumann-Oden DG Formulation: the 3D Case
JO  - Bollettino della Unione matematica italiana
PY  - 2008
SP  - 629
EP  - 643
VL  - 1
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a6/
LA  - en
ID  - BUMI_2008_9_1_3_a6
ER  - 
%0 Journal Article
%A Antonietti, Paola F.
%A Brezzi, Franco
%A Marini, L.Donatella
%T Stabilizations of the Baumann-Oden DG Formulation: the 3D Case
%J Bollettino della Unione matematica italiana
%D 2008
%P 629-643
%V 1
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a6/
%G en
%F BUMI_2008_9_1_3_a6
Antonietti, Paola F.; Brezzi, Franco; Marini, L.Donatella. Stabilizations of the Baumann-Oden DG Formulation: the 3D Case. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 629-643. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a6/

[1] P. F. Antonietti - F. Brezzi - L.D. Marini, Bubble Stabilizations of DG methods, submitted to Comput. Methods Appl. Mech. Engrg. | DOI | MR

[2] D.N. Arnold - F. Brezzi - B. Cockburn - L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39 (2002), 1749-1779. | DOI | MR | Zbl

[3] C. Baiocchi - F. Brezzi - L.P. Franca, Virtual bubbles and GaLS. Comput. Methods Appl. Mech. Engrg., 105 (1993), 125-141. | DOI | MR | Zbl

[4] C. E. Baumann - J. T. Oden, A discontinuous hp finite element method for the Navier-Stokes equations, 10th. International Conference on Finite Element in Fluids, 1998. | Zbl

[5] C. E. Baumann - J. T. Oden, A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 175 (1999), 311-341. | DOI | MR | Zbl

[6] F. Brezzi - M.O. Bristeau - L.P. Franca - M. Mallet - G. Rogé, A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Methods Appl. Mech. Engrg., 96 (1992), 117-129. | DOI | MR

[7] F. Brezzi - B. Cockburn - L.D. Marini - E. Süli, Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Engrg., 195, No. 25-28 (2006), 3293-3310. | DOI | MR

[8] F. Brezzi - L. P. Franca - T. J. R. Hughes - A. Russo, $b\int_g$. Comput. Methods Appl. Mech. Engrg., 142 (1997), 353-360. g | DOI | MR

[9] F. Brezzi - L.D. Marini, Error estimates for the three-field formulation with bubble stabilization. Math. Comp., 70 (2001), 911-934. | DOI | MR | Zbl

[10] F. Brezzi - L.D. Marini, Bubble stabilization of discontinuous Galerkin methods, in W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, Y. Vassilevski (Eds.): Proceedings of International Conference on the occasion of the 60th birthday of Y.A. Kuznetsov Institute of Numerical Mathematics of the Russian Academy of Sciences. Moskow, (2006) 25-36.

[11] F. Brezzi - L.D. Marini - A. Russo, Applications of Pseudo Residual-Free Bubbles to the Stabilization of Convection-Diffusion Problems. Comput. Methods Appl. Mech. Engrg., 166 (1998), 51-63. | DOI | MR | Zbl

[12] F. Brezzi - L.D. Marini - A. Russo, On the choice of a Stabilizing Subgrid for Convection-Diffusion Problems. Comput. Methods Appl. Mech. Engrg., 194 (2005), 127-148. | DOI | MR | Zbl

[13] F. Brezzi - L.D. Marini - E. Süli, Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Models Methods Appl. Sci., 14, No.12 (2004), 1893-1903. | DOI | MR | Zbl

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

[15] 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 (1982), 199-259. | DOI | MR | Zbl

[16] A. Buffa, Error estimate for a stabilised domain decomposition method with nonmatching grids, Numer. Math., 90 (2002), 617-640. | DOI | MR | Zbl

[17] E. Burman - B. Stamm, Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment, C. R. Acad. Sci. Paris, Sr. I., 346, No 1-2 (2008), 103-106. | DOI | MR | Zbl

[18] E. Burman - B. Stamm, Low order discontinuous Galerkin methods for second order elliptic problems, EPFL-IACS report 04.2007, submitted to SIAM J. Numer. Anal. | DOI | MR

[19] E. Burman - B. Stamm, Minimal stabilization of discontinuous Galerkin finite element methods for hyperbolic problems, J. Sci. Comput., 33, No. 2 (2007), 183- 208. | DOI | MR | Zbl

[20] B. Cockburn, Discontinuous Galerkin methods, Zeitschirift fur Angewandte Mathematik und Mechanick, 83, No.11 (2003), 731-754. | DOI | MR | Zbl

[21] L. P. Franca - A. Russo, Deriving upwinding, mass lumping and selective reduced integration by RFB. Appl. Math. Letters, 9 (1996), 83-88. | DOI | MR | Zbl

[22] P. Lesaint - P.-A. Raviart, On a finite element method for solving the neutron transport equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations (Ed. C.A. deBoor), Academic Press (New York, 1974), 89-123. | MR

[23] J. T. Oden - I. Babuska - C. E. Baumann, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., 146 (1998), 491-519. | DOI | MR | Zbl

[24] B. Rivière - M. F. Wheeler - V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Comp. Geo., 3 (1999), 337-360. | DOI | MR | Zbl