Maio 2014 vol. 1 num. 1 - 10th World Congress on Computational Mechanics
Full Article - Open Access.
NUMERICAL ANALYSIS OF A LOCALLY PROJECTED DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC PROBLEMS
In this paper we study a new discontinuousGalerkin method which uses a computational structure compatible with conforming finite element methods, reducing considerably the number of degrees-of-freedom. The Locally Discontinuous but Globally Continuous Galerkin method starts with a discontinuous finite element space and constructs a continuous representation for it by means of local projections. The used technique is similar to that employed in hybridizable methods, and discontinuous solution is recovered by solving local element-wise problems. We present the numerical analysis of the method and numerical results to confirm the predicted convergence rates. Moreover, numerical experiments are conducted in order to evaluate the better performance of this formulation when compared to the continuous or discontinuous Galerkin formulations.
Palavras-chave: discontinuous Galerkin, hybridizable discontinuous Galerkin,
-  S. Adjerid, K. D. Devine, J. E. Flaherty, and L. Krivodonova. A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Computer Methods in Applied Mechanics and Engineering, 191(11-12):1097 – 1112, 2002.
-  D. N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19(4):742–760, 198
-  D. N. Arnold, F. Brezzi, B. Cockburn, and D. Marini. Discontinuous Galerkin methods for elliptic problems. In Discontinuous Galerkin methods (Newport, RI, 1999), volume 11 of Lect. Notes Comput. Sci. Eng., pages 89–101. Springer, Berlin, 2000.
-  D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779, 2001/02.
-  N. C. Arruda, R. C. Almeida, and E. G. D. do Carmo. Discontinuous subgrid formulations for transport problems. ComputerMethods in AppliedMechanics and Engineering, 199(49-52):3227 – 3236, 2010.
-  N. C. Arruda, A. F. D. Loula, and R. C. Almeida. Locally continuous but globally continuous galerkin methods for elliptic problems. (submitted).
-  C. E. Baumann and J. T. Oden. A discontinuous hp finite elementmethod for convectiondiffusion problems. Comput. Methods Appl. Mech. Engrg., 175(3-4):311–341, 1999.
-  F. Brezzi, B. Cockburn, L. D. Marini, and E. Süli. Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Engrg., 195(25-28):3293–3310, 2006.
-  A. Buffa, T. J. R. Hughes, and G. Sangalli. Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems. SIAM J. Numer. Anal., 44(4):1420– 1440 (electronic), 2006.
-  B. Cockburn, B. Dong, and J. Guzmán. A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comp., 77(264):1887–1916, 2008.
-  B. Cockburn, B. Dong, J. Guzmán, M. Restelli, and R. Sacco. A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM J. Sci. Comput., 31(5):3827–3846, 2009.
-  B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of discontinuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems. 47(2):1319–1365, 2009.
-  B. Cockburn, J. Gopalakrishnan, and F.-J. Sayas. A projection-based error analysis of HDG methods. Math. Comp., 79(271):1351–1367, 2010.
-  E. G. D. do Carmo and A. V. C. Duarte. New formulations and numerical analysis of discontinuous Galerkin methods. Comput. Appl. Math., 21(3):661–715, 2002.
-  E. G. Dutra do Carmo and A. V. C. Duarte. A discontinuous finite element-based domain decomposition method. Comput. Methods Appl. Mech. Engrg., 190(8-10):825– 843, 2000.
-  P. Houston, C. Schwab, and E. Süli. Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal., 39(6):2133–2163, 2002.
-  T. J. R. Hughes, G. Scovazzi, P. B. Bochev, and A. Buffa. A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method. Comput. Methods Appl. Mech. Engrg., 195(19-22):2761–2787, 2006.
-  C. Johnson, U. Nävert, and J. Pitkäranta. Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg., 45(1-3):285–312, 1984.
-  C. Johnson and J. Pitkäranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp., 46(173):1–26, 1986.
-  S. Kaya and B. Rivière. A discontinuous subgrid eddy viscosity method for the timedependent Navier-Stokes equations. SIAM J. Numer. Anal., 43(4):1572–1595 (electronic), 2005.
-  W. Klieber and B. Rivière. Adaptive simulations of two-phase flow by discontinuous galerkin methods. Comput. Methods Appl. Mech. Engrg., 196:404–419, 2006.
-  P. Lesaint and P.-A. Raviart. On a finite element method for solving the neutron transport equation. In Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), pages 89– 123. Publication No. 33. Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974.
-  N. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous galerkin method for nonlinear convection-diffusion equations. Journal of Computational Physics, 228(23):8841 – 8855, 2009.
-  W. H. Reed and T. R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
-  B. Rivière. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: theory and implementation. SIAM, 2008.
-  F. A. Rochinha, G. B. Alvarez, E. G. D. do Carmo, and A. F. D. Loula. A locally discontinuous enriched finite element formulation for acoustics. Comm. Numer. Methods Engrg., 23(6):623–637, 2007.
-  S. Sun and M. F. Wheeler. Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal., 43(1):195–219, 2005.
Arruda, N. C. B.; Loula, A. F. D.; Almeida, R. C.; "NUMERICAL ANALYSIS OF A LOCALLY PROJECTED DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC PROBLEMS", p. 2674-2689 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1].
São Paulo: Blucher,
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-18968
últimos 30 dias | último ano | desde a publicação