Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem

Computer Methods in Applied Mechanics and Engineering

Volumn 195, Issue 25-28, 2006, Pages 3331-3346

  Adjerid, Slimane ^a   Massey, Thomas C ^b  

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

^b NAVAL RESEARCH LABORATORY   (United States)

Author keywords

Conservation laws; Discontinuous finite element methods; Hyperbolic differential equations; Superconvergence

Indexed keywords

COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS;

CONSERVATION LAWS; DISCONTINUOUS FINITE ELEMENT METHODS; DISCONTINUOUS GALERKIN SOLUTIONS; HYPERBOLIC DIFFERENTIAL EQUATIONS; NONLINEAR SCALAR HYPERBOLIC;

GALERKIN METHODS;

COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; GALERKIN METHODS; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS;

EID: 33645910891     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2005.06.017     Document Type: Article

References (25)

1. 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
2. S. Adjerid, C. Ermira, T. Weinhart, Superconvergence and error estimation of discontinuous finite element error for transient hyperbolic problems, in preparation.
3. Adjerid S., and Klauser A. Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems. J. Sci. Comput. (2005) 5-24
4. 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
5. Bassi F., and Rebay S. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267-279
6. Baumann C.E., and Oden J.T. A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311-341
7. Bey K.S., and Oden J.T. Hp-version discontinuous Galerkin method for hyperbolic conservation laws. Comput. Methods Appl. Mech. Engrg. 133 (1996) 259-286
8. Bey K.S., Oden J.T., and Patra A. hp-version discontinuous Galerkin method for hyperbolic conservation laws: a parallel strategy. Int. J. Numer. Methods Engrg. 38 (1995) 3889-3908
9. Bey K.S., Oden J.T., and Patra A. A parallel hp-adaptive discontinuous Galerkin method for hyperbolic conservation laws. Appl. Numer. Math. 20 (1996) 321-386
10. Biswas R., Devine K., and Flaherty J.E. Parallel adaptive finite element methods for conservation laws. Appl. Numer. Math. 14 (1994) 255-284
11. Cockburn B. Discontinuous Galerkin methods for convection dominated problems. In: Barth T., and Deconink H. (Eds). High-order Methods for Computational Physics. Lecture Notes in Computational Science and Engineering vol. 9 (1999), Springer, Berlin 69-224
12. Cockburn B., and Gremaud P.A. Error estimates for finite element methods for nonlinear conservation laws. SIAM J. Numer. Anal. 33 (1996) 522-554
13. In: Cockburn B., Karniadakis G.E., and Shu C.W. (Eds). Discontinuous Galerkin methods theory, computation and applications. Lectures Notes in Computational Science and Engineering vol. 11 (2000), Springer, Berlin
14. Cockburn B., Lin S.Y., and Shu C.W. TVB Runge-Kutta local projection discontinuous Galerkin methods of scalar conservation laws III: one dimensional systems. J. Comput. Phys. 84 (1989) 90-113
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. Devine K.D., and Flaherty J.E. Parallel adaptive hp-refinement techniques for conservation laws. Comput. Methods Appl. Mech. Engrg. 20 (1996) 367-386
17. Ericksson K., and Johnson C. Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28 (1991) 12-23
18. ∞. SIAM J. Numer. Anal. 32 (1995) 706-740
19. L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence, 1991.
20. Flaherty J.E., Loy R., Shephard M.S., Szymanski B.K., Teresco J.D., and Ziantz L.H. Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws. J. Parallel Distributed Comput. 47 (1997) 139-152
21. Jaffré J., Johnson C., and Szepessy A. Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Methods Appl. Sci. 5 (1995) 367-386
22. Krivodonova L., and Flaherty J.E. Error estimation for discontinuous Galerkin solutions of two-dimensional hyperbolic problems. Challenges in computational mathematics (Pohang 2001). Adv. Comput. Math. 19 (2003) 57-71
23. Krivodonova L., Xin J., Remacle J.-F., Chevaugeon N., and Flaherty J. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48 (2004) 323-338
24. W.H. Reed, T.R. Hill, Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, 1973.
25. Wheeler M.F. An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152-161