Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems

Computer Methods in Applied Mechanics and Engineering

Volumn 198, Issue 37-40, 2009, Pages 3113-3129

  Adjerid, Slimane ^a   Weinhart, Thomas ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

A posteriori error estimation; Discontinuous Galerkin method; Linear hyperbolic systems; Superconvergence

Indexed keywords

A POSTERIORI ERROR ESTIMATION; COMPUTATIONAL RESULTS; DISCONTINUOUS GALERKIN; DISCONTINUOUS GALERKIN METHOD; ERROR ESTIMATIONS; FINITE ELEMENT; FIRST-ORDER; HYPERBOLIC SYSTEM; LEADING TERMS; LEGENDRE POLYNOMIALS; LINEAR COMBINATIONS; LINEAR HYPERBOLIC SYSTEMS; LOCAL ERROR; LOCAL PROBLEMS; SUPERCONVERGENCE; THREE DIMENSIONS;

ASYMPTOTIC ANALYSIS; ERROR ANALYSIS; ESTIMATION; PARTIAL DIFFERENTIAL EQUATIONS;

GALERKIN METHODS;

EID: 67949124645     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2009.05.016     Document Type: Article

References (33)

1. In: Abramowitz M., and Stegun I.A. (Eds). Handbook of Mathematical Functions (1965), Dover, New York
2. Adjerid S., and Baccouch M. The discontinuous Galerkin method for two-dimensional hyperbolic problems Part I: Superconvergence error analysis. J. Sci. Comput. 33 (2007) 75-113
3. Adjerid S., and Baccouch M. The discontinuous Galerkin method for two-dimensional hyperbolic problems Part II: A Posteriori error estimation. J. Sci. Comput. 38 (2009) 15-49
4. Adjerid S., Devine K.D., Flaherty J.E., and Krivodonova L. A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1097-1112
5. Adjerid S., and Klauser A. Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems. J. Sci. Comput. 22 (2005) 5-24
6. Adjerid S., and Massey T.C. A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 5877-5897
7. Adjerid S., and Massey T.C. Superconvergence of discontinuous finite element solutions for nonlinear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3331-3346
8. Ainsworth M., and Oden J.T. A Posteriori Error Estimation in Finite Element Analysis (2000), John Wiley, New York
9. K. Bottcher, R. Rannacher, Adaptive error control in solving ordinary differential equations by the discontinuous Galerkin method. Tech. report, University of Heidelberg, 1996.
10. Castillo P. A superconvergence result for discontinuous Galerkin methods applied to elliptic problems. Comput. Methods Appl. Mech. Engrg. 192 (2003) 4675-4685
11. Celiker F., and Cockburn B. Superconvergence of the numerical traces for discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension. Math. Comput. 76 (2007) 67-96
12. B. Cockburn, A simple introduction to error estimation for nonlinear hyperbolic conservation laws, in: Proceedings of the 1998 EPSRC Summer School in Numerical Analysis, SSCM, Graduate Student's Guide for Numerical Analysis, vol. 26, 1999, Springer, Berlin, pp. 1-46.
13. Cockburn B., and Gremaud P.A. Error estimates for finite element methods for nonlinear conservation laws. SIAM J. Numer. Anal. 33 (1996) 522-554
14. In: Cockburn B., Karniadakis G.E., and Shu C.W. (Eds). Discontinuous Galerkin Methods Theory Computation and Applications. Lecture Notes in Computational Science and Engineering vol. 11 (2000), Springer, Berlin
15. Cockburn B., and Shu C.W. TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws II: general framework. Math. Comput. 52 (1989) 411-435
16. Cockburn B., and Shu C.W. The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2240-2463
17. Delfour M., Hager W., and Trochu F. Discontinuous Galerkin methods for ordinary differential equation. Math. Comput. 154 (1981) 455-473
18. Hartmann R., and Houston P. Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservations laws. SIAM J. Sci. Comput. 24 (2002) 979-1004
19. Houston P., Mackenzie J., Süli E., and Warnecke G. A posteriori error analysis for numerical approximations of Friedrichs systems. Numerische Mathematik 82 (1999) 433-470
20. Houston P., Schötzau D., and Wihler T. Energy norm a posteriori error estimation of hp-adaptive discontinuous galerkin methods for elliptic problems. Math. Models Methods Appl. Sci. 17 (2007) 33-62
21. Johnson C. Adaptive finite element methods for diffusion and convection problems. Comput. Methods Appl. Mech. Engrg. 82 (1990) 301-322
22. Johnson C., and Hansbo J. Adaptive finite element methods in computational mechanics. Comput. Methods Appl. Mech. Engrg. 101 (1992) 143-181
23. Johnson C., and Szepessy A. Adaptive finite element methods for conservation laws. Commun. Pure Appl. Math. 48 (1995) 199-243
24. O. Karakashian, C. Makridakis, A space-time finite element method for the nonlinear Shrödinger equation: the discontinuous Galerkin method. Preprint #96-9, Department of Mathematics, University of Crete, 71409 Heraklion-Crete, Greece, 1996.
25. Krivodonova L., and Flaherty J.E. Error estimation for discontinuous Galerkin solutions of two-dimensional hyperbolic problems. Adv. Comput. Math. 19 (2003) 57-71
26. Larson M., and Barth T. A posteriori error estimation for adaptive discontinuous Galerkin approximation of hyperbolic systems. In: Cockburn B., Karniadakis G.E., and Shu C.W. (Eds). Proceedings of the International Symposium on Discontinuous Galerkin Methods Theory, Computation and Applications (2000), Springer, Berlin
27. Lesaint P., and Raviart P. On a finite element method for solving the neutron transport equations. In: de Boor C. (Ed). Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Academic Press, New York 89-145
28. Riviere B., and Wheeler M. A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Comput. Math. Appl. 46 (2003) 143-163
29. Schötzau D., and Schwab C. An hp a-priori error analysis of the DG time-stepping method for initial value problems. Calcolo 37 (2000) 207-232
30. Schötzau D., and Schwab C. Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method. SIAM J. Numer. Anal. 38 (2000) 837-875
31. Süli E. A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems. In: Kröner D., Ohlberger M., and Rhode C. (Eds). An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Lecture Notes in Computational Sciences and Engineering vol. 5 (1999), Springer, Berlin
32. Süli E., and Houston P. Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. In: Duff I., and Watson G. (Eds). State of the Art in Numerical Analysis (1997), Oxford University Press, Oxford
33. Verfurth R. A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement Techniques (1996), Teubner-Wiley, Teubner