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Hybridizable Discontinuous Galerkin Methods

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Spectral and High Order Methods for Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 76))

Abstract

We present an overview of recent developments of HDG methods for numerically solving partial differential equations in fluid mechanics.

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References

  1. D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779, 2001

    Article  MathSciNet  Google Scholar 

  2. P. Bastian and B. Rivière. Superconvergence and H(div) projection for discontinuous Galerkin methods. Int. J. Numer. Methods Fluids, 42:1043–1057, 2003

    Article  MATH  Google Scholar 

  3. P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal., 38(5):1676–1706, 2001

    Article  Google Scholar 

  4. B. Cockburn, B. Dong, and J. Guzmán. A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comp., 77:1887–1916, 2008

    Article  MathSciNet  Google Scholar 

  5. B. Cockburn and J. Gopalakrishnan. A characterization of hybridized mixed methods for second order elliptic problems. SIAM J. Numer. Anal., 42(1):283–301, 2004

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Cockburn, B. Dong, J. Guzmán, M. Restelli, and R. Sacco. An hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM J. Sci. Comput., 31(5):3827–3846, 2009

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Cockburn and J. Gopalakrishnan. The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J. Numer. Anal., 47:1092–1125, 2009

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Cockburn, J. Gopalakrishnan, and J. Guzmán. A new elasticity element made for enforcing weak stress symmetry. Math. Comp., 79:1331–1349, 2009

    Article  Google Scholar 

  9. B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second-order elliptic problems. SIAM J. Numer. Anal., 47:1319–1365, 2009

    Article  MATH  MathSciNet  Google Scholar 

  10. B. Cockburn, J. Gopalakrishnan, and F.-J. Sayas. A projection-based error analysis of HDG methods. Math. Comp., 79:1351–1367, 2010

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Cockburn, J. Gopalakrishnan, N. C. Nguyen, J. Peraire, and F.-J. Sayas. Analysis of an HDG method for Stokes flow. Math. Comp. DOI: 10.1090/S0025-5718-2010-02410-X

    Google Scholar 

  12. B. Cockburn, J. Guzmán, S.-C. Soon, and H. K. Stolarski. An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems. SIAM J. Numer. Anal., 47(4):2686–2707, 2009

    Article  MATH  MathSciNet  Google Scholar 

  13. B. Cockburn, J. Guzmán, and H. Wang. Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comp., 78:1–24, 2009

    Article  MathSciNet  Google Scholar 

  14. B. Cockburn, G. Kanschat, and D. Schötzau. A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comp., 74:1067–1095, 2005

    Article  MATH  MathSciNet  Google Scholar 

  15. B. Cockburn and C. W. Shu. The local discontinuous Galerkin method for convection-diffusion systems. SIAM J. Numer. Anal., 35:2440–2463, 1998

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Fortin, and R. Glowinski. Augmented Lagrangian methods, volume 15 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1983. Applications to the numerical solution of boundary value problems, Translated from the French by B. Hunt and D. C. Spicer

    Google Scholar 

  17. S. Güzey, B. Cockburn, and H. Stolarski. The embedded discontinuous Galerkin methods: Application to linear shells problems, Int. J. Numer. Methods Eng., 70:757–790, 2007

    Article  MATH  Google Scholar 

  18. J.-C. Nédélec. Mixed finite elements in R 3. Numer. Math., 35:315–341, 1980

    Article  MATH  MathSciNet  Google Scholar 

  19. J.-C. Nédélec. A new family of mixed finite elements in R 3. Numer. Math., 50:57–81, 1986

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations. J. Comput. Phys., 228:8841–8855, 2009

    Article  MATH  MathSciNet  Google Scholar 

  22. N. C. Nguyen, J. Peraire, and B. Cockburn. A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Engrg., 199:582–597, 2010

    Article  Google Scholar 

  23. N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. Submitted to Journal of Computational Physics, 2009.

    Google Scholar 

  24. N. C. Nguyen, J. Peraire, and B. Cockburn. A hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. In Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, Florida, January 2010. AIAA-2010-362

    Google Scholar 

  25. J. Peraire, N. C. Nguyen, and B. Cockburn. A hybridizable discontinuous Galerkin method for the compressible Euler and Navier–Stokes Equations. In Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, Florida, January 2010. AIAA-2010-363

    Google Scholar 

  26. P. O. Persson and J. Peraire. Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier–Stokes equations SIAM J. Sci. Comput., 30(6):2709–2733, 2008

    Google Scholar 

  27. S.-C. Soon, B. Cockburn, and H. K. Stolarski. A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Eng., 80(8):1058–1092, 2009

    Article  MATH  MathSciNet  Google Scholar 

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Nguyen, N.C., Peraire, J., Cockburn, B. (2011). Hybridizable Discontinuous Galerkin Methods. In: Hesthaven, J., Rønquist, E. (eds) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 76. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15337-2_4

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