×

On the stability of continuous-discontinuous Galerkin methods for advection-diffusion-reaction problems. (English) Zbl 1282.65131

Summary: We consider a finite element method which couples the continuous Galerkin method away from internal and boundary layers with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. The stability properties of the coupled method are illustrated with a numerical experiment.

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Arnold, DN; Brezzi, F; Cockburn, B; Marini, LD, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779, (2001) · Zbl 1008.65080
[2] Ayuso, B; Marini, LD, Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 47, 1391-1420, (2009) · Zbl 1205.65308
[3] Bardos, C; Rauch, J, Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Am. Math. Soc., 270, 377-408, (1982) · Zbl 0485.35010
[4] Becker, R., Burman, E., Hansbo, P., Larson, M.: A Reduced P1-Discontinuous Galerkin Method. Tech. rep., EPFL (2004) (EPFL-IACS report 05.2004)
[5] Buffa, A; Hughes, TJR; Sangalli, G, Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems, SIAM J. Numer. Anal., 44, 1420-1440, (2006) · Zbl 1153.76038
[6] Cangiani, A., Chapman, J., Georgoulis, E.H., Jensen, M.: On local super-penalization of interior penalty Galerkin methods (2012) (submitted jounral article) · Zbl 1282.65131
[7] Cangiani, A., Georgoulis, E. H., Jensen, M.: Continuous and discontinuous finite element methods for convection-diffusion problems: A comparison. In: International Conference on Boundary and Interior Layers, Göttingen, July (2006) · Zbl 1200.65093
[8] Cockburn, B; Dong, B; Guzmán, J; Restelli, M; Sacco, R, A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems, SIAM J. Sci. Comput., 31, 3827-3846, (2009) · Zbl 1200.65093
[9] Cockburn, B., Karniadakis, G., Shu, C.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G., Shu, C. (eds.) Discontinuous Galerkin Methods: Theory, Computation, and Applications, vol. 11 of Lecture Notes in Computational Science and Engineering. Springer, Berlin (2000) · Zbl 0989.76045
[10] Cockburn, B; Shu, C-W, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463, (1998) · Zbl 0927.65118
[11] Dawson, C; Proft, J, Coupling of continuous and discontinuous Galerkin methods for transport problems, Comput. Methods Appl. Mech. Eng., 191, 3213-3231, (2002) · Zbl 1101.76355
[12] Devloo, PRB; Forti, T; Gomes, SM, A combined continuous-discontinuous finite element method for convection-diffusion problems, Latin Am. J. Solids Struct., 2, 229-246, (2007)
[13] Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004) · Zbl 1059.65103
[14] Nguyen, NC; Peraire, J; Cockburn, B, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228, 3232-3254, (2009) · Zbl 1187.65110
[15] Perugia, I; Schötzau, D, On the coupling of local discontinuous Galerkin and conforming finite element methods, J. Sci. Comput., 16, 411-433, (2001) · Zbl 0995.65117
[16] Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion and Flow Problems, 2nd edn. Springer, Berlin (2008) · Zbl 1155.65087
[17] Scott, LR; Zhang, S, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., 54, 483-493, (1990) · Zbl 0696.65007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.