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Minimal Stabilization for Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems

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Abstract

We consider a discontinuous Galerkin finite element method for the advection–reaction equation in two space–dimensions. For polynomial approximation spaces of degree greater than or equal to two on triangles we propose a method where stability is obtained by a penalization of only the upper portion of the polynomial spectrum of the jump of the solution over element edges. We prove stability in the standard h-weighted graphnorm and obtain optimal order error estimates with respect to mesh-size.

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References

  1. Brezzi, F., Fortin, M.: A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89(3), 457–491 (2001). MR1864427 (2002h:65176)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brezzi, F., Marini, L.D., Süli, E.: Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Models Methods Appl. Sci. 14(12), 1893–1903 (2004). MR2108234 (2005m:65259)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brezzi, F., Cockburn, B., Marini, L.D., Süli, E.: Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Eng. 195(25–28), 3293–3310 (2006). MR2220920

    Article  MATH  Google Scholar 

  4. Burman, E.: A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43(5), 2012–2033 (2005) (electronic). MR2192329

    Article  MATH  MathSciNet  Google Scholar 

  5. Burman, E., Ern, A.: Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations. Math. Comput. 76, 1119–1140 (2007)

    MATH  MathSciNet  Google Scholar 

  6. Burman, E., Hansbo, P.: Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 193(15–16), 1437–1453 (2004). MR2068903 (2005d:65186)

    Article  MATH  MathSciNet  Google Scholar 

  7. Burman, E., Stamm, B.: Discontinuous and continuous finite element methods with interior penalty for hyperbolic problems. Tech. Report, EPFL-IACS report 17 (2005)

  8. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002). MR1930132

    Google Scholar 

  9. Cockburn, B.: Discontinuous Galerkin methods for convection-dominated problems. In: High-Order Methods for Computational Physics. Lect. Notes Comput. Sci. Eng., vol. 9, pp. 69–224. Springer, Berlin (1999). MR1712278 (2000f:76095)

    Google Scholar 

  10. Cockburn, B.: Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws. J. Comput. Appl. Math. 128(1–2), 187–204 (2001). MR1820874 (2001m:65127)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001). MR1873283 (2002i:65099)

    Article  MATH  MathSciNet  Google Scholar 

  12. Douglas, J., Dupont, T.: Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Computing Methods in Applied Sciences, Second Internat. Sympos., Versailles, 1975. Lecture Notes in Phys., vol. 58, pp. 207–216. Springer, Berlin (1976)

    Google Scholar 

  13. Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6(4), 345–390 (1991). MR1154903 (92k:76061)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ern, A., Guermond, J.-L.: Discontinuous Galerkin methods for Friedrichs’ systems. i. General theory. SIAM J. Numer. Anal. 44(2), 753–778 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic, Boston (1994). MR1243179 (94g:00008)

    MATH  Google Scholar 

  16. Guzmán, J.: Local analysis of discontinuous Galerkin methods applied to singularly perturbed problems. J. Numer. Math. 14(1), 41–56 (2006). MR2229818 (2007b:65122)

    Article  MATH  MathSciNet  Google Scholar 

  17. Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002) (electronic). MR1897953 (2003d:65108)

    Article  MATH  MathSciNet  Google Scholar 

  18. Jensen, M.: Discontinuous Galerkin methods for Friedrichs’ systems with irregular solutions. Ph.D. thesis, University of Oxford (2004)

  19. Johnson, C., Pitkäranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46(173), 1–26 (1986). MR815828 (88b:65109)

    Article  MATH  Google Scholar 

  20. Lesaint, P., Raviart, P.-A.: On a finite element method for solving the neutron transport equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations, Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974. Math. Res. Center, Univ. of Wisconsin-Madison, vol. 33, pp. 89–123. Academic, New York (1974). MR0658142 (58 #31918)

    Google Scholar 

  21. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973)

  22. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25. Springer, Berlin (1997). MR1479170 (98m:65007)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Analysis and Scientific Computing, Swiss Institute of Technology, Lausanne, 1015, Switzerland

    E. Burman & B. Stamm

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  1. E. Burman
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  2. B. Stamm
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Correspondence to E. Burman.

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The second author was supported by the Swiss National Science Foundation.

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Burman, E., Stamm, B. Minimal Stabilization for Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems. J Sci Comput 33, 183–208 (2007). https://doi.org/10.1007/s10915-007-9149-5

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  • DOI: https://doi.org/10.1007/s10915-007-9149-5

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