Stabilizations of the Baumann-Oden DG Formulation: the 3D Case

Antonietti, Paola F. ; Brezzi, Franco ; Marini, L.Donatella

Bollettino dell'Unione Matematica Italiana, Tome 1 (2008), p. 629-643 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

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.

Détail

Publié le : 2008-10-01

MR 2455336 | Zbl 1192.65141

Bibliographie

[1] Antonietti, P. F. - Brezzi, F. - Marini, L.D., Bubble Stabilizations of DG methods, submitted to Comput. Methods Appl. Mech. Engrg. | MR 2517937

[2] Arnold, D.N. - Brezzi, F. - Cockburn, B. - Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39 (2002), 1749-1779. | MR 1885715 | Zbl 1008.65080

[3] Baiocchi, C. - Brezzi, F. - Franca, L.P., Virtual bubbles and GaLS. Comput. Methods Appl. Mech. Engrg., 105 (1993), 125-141. | MR 1222297 | Zbl 0772.76033

[4] Baumann, C. E. - Oden, J. T., A discontinuous hp finite element method for the Navier-Stokes equations, 10th. International Conference on Finite Element in Fluids, 1998. | Zbl 0985.76048

[5] Baumann, C. E. - Oden, J. T., A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 175 (1999), 311-341. | MR 1702201 | Zbl 0924.76051

[6] Brezzi, F. - Bristeau, M.O. - Franca, L.P. - Mallet, M. - Rogé, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Methods Appl. Mech. Engrg., 96 (1992), 117-129. | MR 1159592

[7] Brezzi, F. - Cockburn, B. - Marini, L.D. - Süli, E., Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Engrg., 195, No. 25-28 (2006), 3293-3310. | MR 2220920

[8] Brezzi, F. - Franca, L. P. - Hughes, T. J. R. - Russo, A., $b\int_g$. Comput. Methods Appl. Mech. Engrg., 142 (1997), 353-360. g | MR 1456019

[9] Brezzi, F. - Marini, L.D., Error estimates for the three-field formulation with bubble stabilization. Math. Comp., 70 (2001), 911-934. | MR 1826573 | Zbl 0970.65118

[10] Brezzi, F. - Marini, L.D., 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] Brezzi, F. - Marini, L.D. - Russo, A., Applications of Pseudo Residual-Free Bubbles to the Stabilization of Convection-Diffusion Problems. Comput. Methods Appl. Mech. Engrg., 166 (1998), 51-63. | MR 1660129 | Zbl 0932.65113

[12] Brezzi, F. - Marini, L.D. - Russo, A., On the choice of a Stabilizing Subgrid for Convection-Diffusion Problems. Comput. Methods Appl. Mech. Engrg., 194 (2005), 127-148. | MR 2105157 | Zbl 1143.65390

[13] Brezzi, F. - Marini, L.D. - Süli, E., Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Models Methods Appl. Sci., 14, No.12 (2004), 1893-1903. | MR 2108234 | Zbl 1070.65117

[14] Brezzi, F. - Russo, A., Choosing bubbles for advection-diffusion problems. Math. Models Meth. Appl. Sci., 4 (1994), 571-587. | MR 1291139 | Zbl 0819.65128

[15] Brooks, A. N. - Hughes, T. J. R., 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. | MR 679322 | Zbl 0497.76041

[16] Buffa, A., Error estimate for a stabilised domain decomposition method with nonmatching grids, Numer. Math., 90 (2002), 617-640. | MR 1888832 | Zbl 0996.65112

[17] Burman, E. - Stamm, B., 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. | MR 2385065 | Zbl 1133.65096

[18] Burman, E. - Stamm, B., Low order discontinuous Galerkin methods for second order elliptic problems, EPFL-IACS report 04.2007, submitted to SIAM J. Numer. Anal. | MR 2475950

[19] Burman, E. - Stamm, B., Minimal stabilization of discontinuous Galerkin finite element methods for hyperbolic problems, J. Sci. Comput., 33, No. 2 (2007), 183- 208. | MR 2342594 | Zbl 1132.65084

[20] Cockburn, B., Discontinuous Galerkin methods, Zeitschirift fur Angewandte Mathematik und Mechanick, 83, No.11 (2003), 731-754. | MR 2015264 | Zbl 1036.65079

[21] Franca, L. P. - Russo, A., Deriving upwinding, mass lumping and selective reduced integration by RFB. Appl. Math. Letters, 9 (1996), 83-88. | MR 1415477 | Zbl 0903.65082

[22] Lesaint, P. - Raviart, P.-A., 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 658142

[23] Oden, J. T. - Babuska, I. - Baumann, C. E., A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., 146 (1998), 491-519. | MR 1654911 | Zbl 0926.65109

[24] Rivière, B. - Wheeler, M. F. - Girault, V., Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Comp. Geo., 3 (1999), 337-360. | MR 1750076 | Zbl 0951.65108