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On the Stability of Continuous–Discontinuous Galerkin Methods for Advection–Diffusion–Reaction Problems

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Abstract

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.

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References

  1. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2001)

    Article  MathSciNet  Google Scholar 

  2. Ayuso, B., Marini, L.D.: Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47, 1391–1420 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bardos, C., Rauch, J.: Maximal positive boundary value problems as limits of singular perturbation problems. Trans. Am. Math. Soc. 270, 377–408 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  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, T.J.R., Sangalli, G.: Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems. SIAM J. Numer. Anal. 44, 1420–1440 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cangiani, A., Chapman, J., Georgoulis, E.H., Jensen, M.: On local super-penalization of interior penalty Galerkin methods (2012) (submitted jounral article)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dawson, C., Proft, J.: Coupling of continuous and discontinuous Galerkin methods for transport problems. Comput. Methods Appl. Mech. Eng. 191, 3213–3231 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Devloo, P.R.B., Forti, T., Gomes, S.M.: A combined continuous-discontinuous finite element method for convection-diffusion problems. Latin Am. J. Solids Struct. 2, 229–246 (2007)

    Google Scholar 

  13. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004)

    Book  MATH  Google Scholar 

  14. Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations. J. Comput. Phys. 228, 3232–3254 (2009)

    Article  MathSciNet  Google Scholar 

  15. Perugia, I., Schötzau, D.: On the coupling of local discontinuous Galerkin and conforming finite element methods. J. Sci. Comput. 16, 411–433 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  17. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We gratefully thank the Archimedes Center for Modeling, Analysis and Computation in Crete for hosting the authors during the preparation of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

    Andrea Cangiani & Emmanuil Georgoulis

  2. Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK

    John Chapman & Max Jensen

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  1. Andrea Cangiani
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  2. John Chapman
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  3. Emmanuil Georgoulis
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  4. Max Jensen
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Correspondence to Max Jensen.

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Cangiani, A., Chapman, J., Georgoulis, E. et al. On the Stability of Continuous–Discontinuous Galerkin Methods for Advection–Diffusion–Reaction Problems. J Sci Comput 57, 313–330 (2013). https://doi.org/10.1007/s10915-013-9707-y

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  • DOI: https://doi.org/10.1007/s10915-013-9707-y

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