Minimal stabilization for discontinuous Galerkin finite element methods for hyperbolic problems

Journal of Scientific Computing

Volumn 33, Issue 2, 2007, Pages 183-208

  Burman, E ^a   Stamm, B ^a  

^a EPFL   (Switzerland)

Author keywords

Advection reaction equation; Discontinuous Galerkin h FEM; Interior penalty; Local mass conservation

Indexed keywords

GALERKIN FINITE ELEMENT METHODS; HYPERBOLIC PROBLEMS; INTERIOR PENALTY; LOCAL MASS CONSERVATION;

ERROR ANALYSIS; GALERKIN METHODS; MESH GENERATION; POLYNOMIAL APPROXIMATION; PROBLEM SOLVING;

FINITE ELEMENT METHOD;

EID: 34748829146     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-007-9149-5     Document Type: Article

References (22)

1. Brezzi, F., Fortin, M.: A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89(3), 457-491 (2001). MR1864427 (2002h:65176)
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)
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
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
5. Burman, E., Ern, A.: Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations. Math. Comput. 76, 1119-1140 (2007)
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)
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
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)
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)
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)
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)
13. Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6(4), 345-390 (1991). MR1154903 (92k:76061)
14. Ern, A., Guermond, J.-L.: Discontinuous Galerkin methods for Friedrichs' systems. i. General theory. SIAM J. Numer. Anal. 44(2), 753-778 (2006)
15. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic, Boston (1994). MR1243179 (94g:00008)
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)
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)
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)
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)
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)