Services on Demand
Journal
Article
Spanish (pdf)
Article in xml format
Article references
Automatic translation
Send this article by e-mailIndicators
-
Cited by SciELO -
Access statistics
Related links
-
Similars in
SciELO
Share
Permalink
Revista de Matemática Teoría y Aplicaciones
Print version ISSN 1409-2433
Rev. Mat vol.19 n.2 San José Jul. 2012
Implementación del método LDG para mallas no estructuradas en 3D
Implementation of ldg method for 3d unstructured meshes
Implementation of ldg method for 3d unstructured meshes
Resumen
En este artículo se describe una implementación del método “Local Discontinuous Galerkin” (LDG) aplicado a problemas elípticos en 3D. Se discute la implementación de los principales operadores. En particular el uso de aproximaciones de alto orden y de mallas no estructuradas. Estructuras de datos eficientes que permiten un rápido ensamblado del sistema lineal en su formulación mixta son descritas en detalle.
Palabras clave: Métodos de elemento finito discontinuos, aproximaciones de alto orden, mallas no estructuradas, programación orientada a objetos.
Abstract
This paper describes an implementation of the Local Discontinuous Galerkin method (LDG) applied to elliptic problems in 3D. The implementation of the major operators is discussed. In particular the use of higher-order approximations and unstructured meshes. Efficient data structures that allow fast assembly of the linear system in the mixed formulation are described in detail.
Keywords: Discontinuous finite element methods, high-order approximations, unstructured meshes, object-oriented programming.
Mathematics Subject Classification: 65K05, 65N30, 65N55.
En este artículo se describe una implementación del método “Local Discontinuous Galerkin” (LDG) aplicado a problemas elípticos en 3D. Se discute la implementación de los principales operadores. En particular el uso de aproximaciones de alto orden y de mallas no estructuradas. Estructuras de datos eficientes que permiten un rápido ensamblado del sistema lineal en su formulación mixta son descritas en detalle.
Palabras clave: Métodos de elemento finito discontinuos, aproximaciones de alto orden, mallas no estructuradas, programación orientada a objetos.
Abstract
This paper describes an implementation of the Local Discontinuous Galerkin method (LDG) applied to elliptic problems in 3D. The implementation of the major operators is discussed. In particular the use of higher-order approximations and unstructured meshes. Efficient data structures that allow fast assembly of the linear system in the mixed formulation are described in detail.
Keywords: Discontinuous finite element methods, high-order approximations, unstructured meshes, object-oriented programming.
Mathematics Subject Classification: 65K05, 65N30, 65N55.
Ver contenido disponible en pdf
Referencias
[1] Anderson, M.; Bai, Z.; Bischof, C:; Blacford, S.; Demmel, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKennes, A.; Sorensen, D. (1999) LAPACK User’s Guide, 3rd edition. Society for Industrial and Applied Mathematics, Philadelphia PA. [ Links ]
[2] Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, D. (2002) “A unified analysis for discontinuous Galerkin methods for elliptic problems”, SIAM J. Num. Anal. 39(5):1749–1779. [ Links ]
[3] W. Bangerth, R. Hartmann, and G. Kanschat. Deal.II: a general purpose object oriented finite element library. ACM. Trans. Math. Software., 33(4):24:1–24:27, 2007. [ Links ]
[4] Bustinza, R.; Gatica. G.N. (2004) “A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions”, SIAM J. Sci. Comput. 26(1): 152–177. [ Links ]
[5] Castillo, P. (2002) “Performance of discontinuous Galerkin methods for elliptic PDE’s”, SIAM J. Sci. Comput. 24(2): 524–547. [ Links ]
[6] Castillo, P. (2010) “Stencil reduction algorithms for the local discontinuous Galerkin method”, Internat. J. Numer. Methods Engrg. 81: 1475–1491. [ Links ]
7 Castillo, P.; Cockburn, B.; Perugia, I.; Sch¨otzau, D. (2000) “An a priori error analysis of the local discontinuous Galerkin method for elliptic problems”, SIAM J. Num. Anal. 38(5): 1676–1706. [ Links ]
[8] Chow, E.; Heroux, M.A. (1998) “An object-oriented framework for block preconditioning”, ACM Trans. Math. Softw. 24: 159–183. [ Links ]
[9] Cockburn, B.; Shu, C.W. (1998) The local discontinuous Galerkin method for time-dependent convection-diffusion systems”, SIAM J. Num. Anal. 35: 2440–2463. [ Links ]
[10] Perugia, I.; Schötzau, D. (2002) “An hp analysis of the local discontinuous Galerkin method for diffusion problems”, J. Scientific Computing. 17: 561–571. [ Links ]
[11] Saad, Y. (2003) Iterative Methods for Sparse Linear Systems, 2nd edition. Society for Industrial and Applied Mathematics, Philadelphia PA. [ Links ].
[12] Tetgen, H.S. (2004) “A quality tetrahedral mesh generator and three dimensional Delaunay triangulator, v.1.3, user’s manual”. Technical Report 9, Weierstrass Institute for Applied Analysis and Stochastics, Berlin. [ Links ]
*Correspondencia a: Filánder A. Sequeira Chavarría. Escuela de Matemática, Universidad de Costa Rica, 2060 San José, Costa Rica. E-Mail: filander.sequeira@ucr.ac.cr
Paul E. Castillo. Departamento de Ciencias Matem´aticas, P.O. Box 9000, Universidad de Puerto Rico, 00681 Mayagüez Puerto Rico. E-Mail: paul.castillo@upr.edu
*Escuela de Matemática, Universidad de Costa Rica, 2060 San José, Costa Rica. E-Mail: filander.sequeira@ucr.ac.cr
†Departamento de Ciencias Matem´aticas, P.O. Box 9000, Universidad de Puerto Rico, 00681 Mayagüez Puerto Rico. E-Mail: paul.castillo@upr.edu
[1] Anderson, M.; Bai, Z.; Bischof, C:; Blacford, S.; Demmel, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKennes, A.; Sorensen, D. (1999) LAPACK User’s Guide, 3rd edition. Society for Industrial and Applied Mathematics, Philadelphia PA. [ Links ]
[2] Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, D. (2002) “A unified analysis for discontinuous Galerkin methods for elliptic problems”, SIAM J. Num. Anal. 39(5):1749–1779. [ Links ]
[3] W. Bangerth, R. Hartmann, and G. Kanschat. Deal.II: a general purpose object oriented finite element library. ACM. Trans. Math. Software., 33(4):24:1–24:27, 2007. [ Links ]
[4] Bustinza, R.; Gatica. G.N. (2004) “A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions”, SIAM J. Sci. Comput. 26(1): 152–177. [ Links ]
[5] Castillo, P. (2002) “Performance of discontinuous Galerkin methods for elliptic PDE’s”, SIAM J. Sci. Comput. 24(2): 524–547. [ Links ]
[6] Castillo, P. (2010) “Stencil reduction algorithms for the local discontinuous Galerkin method”, Internat. J. Numer. Methods Engrg. 81: 1475–1491. [ Links ]
7 Castillo, P.; Cockburn, B.; Perugia, I.; Sch¨otzau, D. (2000) “An a priori error analysis of the local discontinuous Galerkin method for elliptic problems”, SIAM J. Num. Anal. 38(5): 1676–1706. [ Links ]
[8] Chow, E.; Heroux, M.A. (1998) “An object-oriented framework for block preconditioning”, ACM Trans. Math. Softw. 24: 159–183. [ Links ]
[9] Cockburn, B.; Shu, C.W. (1998) The local discontinuous Galerkin method for time-dependent convection-diffusion systems”, SIAM J. Num. Anal. 35: 2440–2463. [ Links ]
[10] Perugia, I.; Schötzau, D. (2002) “An hp analysis of the local discontinuous Galerkin method for diffusion problems”, J. Scientific Computing. 17: 561–571. [ Links ]
[11] Saad, Y. (2003) Iterative Methods for Sparse Linear Systems, 2nd edition. Society for Industrial and Applied Mathematics, Philadelphia PA. [ Links ].
[12] Tetgen, H.S. (2004) “A quality tetrahedral mesh generator and three dimensional Delaunay triangulator, v.1.3, user’s manual”. Technical Report 9, Weierstrass Institute for Applied Analysis and Stochastics, Berlin. [ Links ]
*Correspondencia a: Filánder A. Sequeira Chavarría. Escuela de Matemática, Universidad de Costa Rica, 2060 San José, Costa Rica. E-Mail: filander.sequeira@ucr.ac.cr
Paul E. Castillo. Departamento de Ciencias Matem´aticas, P.O. Box 9000, Universidad de Puerto Rico, 00681 Mayagüez Puerto Rico. E-Mail: paul.castillo@upr.edu
*Escuela de Matemática, Universidad de Costa Rica, 2060 San José, Costa Rica. E-Mail: filander.sequeira@ucr.ac.cr
†Departamento de Ciencias Matem´aticas, P.O. Box 9000, Universidad de Puerto Rico, 00681 Mayagüez Puerto Rico. E-Mail: paul.castillo@upr.edu
Received: 21 Feb 2011; Revised: 12 Jun 2012; Accepted: 18 Jun 2012
