Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A direct coupling of local discontinuous Galerkin and boundary element methods


Authors: Gabriel N. Gatica, Norbert Heuer and Francisco-Javier Sayas
Journal: Math. Comp. 79 (2010), 1369-1394
MSC (2000): Primary 65N30, 65N38, 65N12, 65N15
DOI: https://doi.org/10.1090/S0025-5718-10-02309-4
Published electronically: January 8, 2010
MathSciNet review: 2629997
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The coupling of local discontinuous Galerkin (LDG) and boundary element methods (BEM), which has been developed recently to solve linear and nonlinear exterior transmission problems, employs a mortar-type auxiliary unknown to deal with the weak continuity of the traces at the interface boundary. As a consequence, the main features of LDG and BEM are maintained and hence the coupled approach benefits from the advantages of both methods. In this paper we propose and analyze a simplified procedure that avoids the mortar variable by employing LDG subspaces whose functions are continuous on the coupling boundary. The continuity can be implemented either directly or indirectly via the use of Lagrangian multipliers. In this way, the normal derivative becomes the only boundary unknown, and hence the total number of unknown functions is reduced by two. We prove the stability of the new discrete scheme and derive an a priori error estimate in the energy norm. A numerical example confirming the theoretical result is provided. The analysis is also extended to the case of nonlinear problems and to the coupling with other discontinuous Galerkin methods.


References [Enhancements On Off] (What's this?)

  • 1. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis, vol. 19, 4, pp. 742-760, (1982). MR 664882 (83f:65173)
  • 2. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis, vol. 39, 5, pp. 1749-1779, (2002). MR 1885715 (2002k:65183)
  • 3. I. Babuška and M. Suri, The h-p version of the finite element method with quasiuniform meshes. RAIRO Modélisation Mathématique et Analyse Numérique, vol. 21, pp. 199-238, (1987). MR 896241 (88d:65154)
  • 4. I. Babuška and M. Suri, The optimal convergence rate of the p-version of the finite element method. SIAM Journal on Numerical Analysis, vol. 24, pp. 750-776, (1987). MR 899702 (88k:65102)
  • 5. I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty. SIAM Journal on Numerical Analysis, vol. 10, pp. 863-875, (1973). MR 0345432 (49:10168)
  • 6. A. Bespalov and N. Heuer, The $ hp$-version of the boundary element method with quasi-uniform meshes in three dimensions. ESAIM Mathematical Modelling and Numerical Analysis, vol. 42, 5, pp. 821-849, (2008). MR 2454624 (2009f:65277)
  • 7. R. Bustinza and G.N. Gatica, A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions. SIAM Journal on Scientific Computing, vol. 26, 1, pp. 152-177, (2004). MR 2114338 (2005k:65201)
  • 8. R. Bustinza and G.N. Gatica, A mixed local discontinuous Galerkin method for a class of nonlinear problems in fluid mechanics. Journal of Computational Physics, vol. 207, pp. 427-456, (2005). MR 2144625 (2006a:76069)
  • 9. R. Bustinza, G.N. Gatica and F.-J. Sayas, On the coupling of local discontinuous Galerkin and boundary element methods for nonlinear exterior transmission problems. IMA Journal of Numerical Analysis, vol. 28, 2, pp. 225-244, (2008). MR 2401197 (2009c:65295)
  • 10. R. Bustinza, G.N. Gatica and F.-J. Sayas, A LDG-BEM coupling for a class of nonlinear exterior transmission problems. In Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2005 (A. Bermúdez de Castro, D. Gómez, P. Quintela, and P. Salgado, eds.), pp. 1129-1136, Springer-Verlag, 2006. MR 2303745
  • 11. R. Bustinza, G.N. Gatica and F.-J. Sayas, A look at how LDG and BEM can be coupled. ESAIM Proceedings, vol. 21, pp. 88-97, (2007). MR 2404055 (2009c:65332)
  • 12. B. Cockburn and C. Dawson, Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions. In Proceedings of the 10th Conference on the Mathematics of Finite Elements and Applications, J. R. Whiteman, ed., Elsevier, 2000, pp. 225-238. MR 1801979 (2001j:65142)
  • 13. M. Costabel, Symmetric methods for the coupling of finite elements and boundary elements. In Boundary Elements IX, vol. 1 (C. A. Brebbia et al., eds.), pp. 411-420, Springer, 1987 MR 965328 (89j:65068)
  • 14. M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results. SIAM Journal on Mathematical Analysis, vol. 19, pp. 613-626, (1988). MR 937473 (89h:35090)
  • 15. J. Douglas Jr., T. Dupont, Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods. Lecture notes in Physics, vol. 58, Springer, Berlin, 1976. MR 0440955 (55:13823)
  • 16. M. Feistauer, Mathematical and numerical study of nonlinear problems in fluid mechanics. In Proc. Conf. Equadiff 6, edited by J. Vosmansky and M. Zlámal, Brno 1985, Springer, Berlin, pp. 3-16. MR 877102 (88f:76002)
  • 17. M. Feistauer, On the finite element approximation of a cascade flow problem. Numerische Mathematik, vol. 50, pp. 655-684, (1997). MR 884294 (88h:65205)
  • 18. G.N. Gatica and F.-J. Sayas, An a priori error analysis for the coupling of local discontinuous Galerkin and boundary element methods. Mathematics of Computation, vol. 75, pp. 1675-1696, (2006). MR 2240630 (2007e:65119)
  • 19. P. Grisvard, Singularities in Boundary Value Problems. Recherches in Mathématiques Appliquées, Masson, Paris, 1992. MR 1173209 (93h:35004)
  • 20. H. Han, A new class of variational formulations for the coupling of finite and boundary element methods. Journal of Computational Mathematics, vol. 8, 3, pp. 223-232, (1990). MR 1299224
  • 21. B. Heise, Nonlinear field calculations with multigrid Newton methods. Impact of Computing in Science and Engineering, vol. 5, pp. 75-110, (1993). MR 1223880 (95a:78002)
  • 22. B. Heise, Analysis of a fully discrete finite element method for a nonlinear magnetic field problem. SIAM Journal on Numerical Analysis, vol. 31, 3, pp. 745-759, (1994). MR 1275111 (95i:65156)
  • 23. P. Houston, J. Robson and E. Süli, Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case. IMA Journal of Numerical Analysis, vol. 25, 4, pp. 726-749, (2005). MR 2170521 (2006k:65322)
  • 24. G.C. Hsiao and W. Wendland, Boundary Integral Equations. Applied Mathematical Sciences, vol. 164, Springer-Verlag Berlin Heidelberg, 2008. MR 2441884 (2009i:45001)
  • 25. J.-C. Nédélec, Integral equations with nonintegrable kernels. Integral Equations and Operator Theory, vol. 5, pp. 562-572, (1982). MR 665149 (84i:45011)
  • 26. B. Rivière, M.F. Wheeler and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I. Computational Geosciences, vol. 3, pp. 337-360, (1999). MR 1750076 (2001d:65145)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N38, 65N12, 65N15

Retrieve articles in all journals with MSC (2000): 65N30, 65N38, 65N12, 65N15


Additional Information

Gabriel N. Gatica
Affiliation: CI$^{2}$MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Email: ggatica@ing-mat.udec.cl

Norbert Heuer
Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile,
Email: nheuer@mat.puc.cl

Francisco-Javier Sayas
Affiliation: Departamento de Matemática Aplicada, Centro Politécnico Superior, Universidad de Zaragoza, María de Luna, 3 - 50018 Zaragoza, Spain
Address at time of publication: School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, Minnesota 55455 USA
Email: jsayas@unizar.es

DOI: https://doi.org/10.1090/S0025-5718-10-02309-4
Keywords: Boundary elements, local discontinuous Galerkin method, coupling, error estimates.
Received by editor(s): November 1, 2007
Received by editor(s) in revised form: April 29, 2009
Published electronically: January 8, 2010
Additional Notes: This research was partially supported by FONDAP and BASAL projects CMM, Universidad de Chile, by Centro de Investigación en Ingeniería Matemática (CI$^{2}$MA), Universidad de Concepción, by FONDECYT project no. 1080044, by Spanish FEDER/MCYT Project MTM2007-63204, and by Gobierno de Aragón (Grupo Consolidado PDIE)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society