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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
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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.


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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.