Optimal error estimates for the fully discrete interior penalty DG method for the wave equation

Journal of Scientific Computing

Volumn 40, Issue 1-3, 2009, Pages 257-272

  Grote, Marcus J ^a   Schötzau, Dominik ^b  

^a UNIVERSITY OF BASEL   (Switzerland)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Discontinuous Galerkin methods; Finite element methods; Interior penalty method; Leap frog scheme; Wave equation

Indexed keywords

APRIORI; DISCONTINUOUS GALERKIN METHODS; ENERGY NORM; FINITE DIFFERENCE APPROXIMATIONS; FINITE TIME INTERVALS; FULLY DISCRETE; INTERIOR PENALTY METHOD; LEAP-FROG SCHEME; MAXIMAL ERROR; MESH SIZE; NUMERICAL SCHEME; OPTIMAL ERROR ESTIMATE; POLYNOMIAL DEGREE; SECOND ORDERS; SMOOTH SOLUTIONS; TIME DISCRETIZATION; TIME STEP; TIME-DEPENDENT WAVE EQUATIONS;

ERROR ANALYSIS; FINITE ELEMENT METHOD; WAVE EQUATIONS;

GALERKIN METHODS;

EID: 67349191482     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-008-9247-z     Document Type: Article

References (30)

1. M. Ainsworth P. Monk W. Muniz 2006 Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation J. Sci. Comput. 27 5 40 (Pubitemid 43876927)
2. D.N. Arnold F. Brezzi B. Cockburn L.D. Marini 2002 Unified analysis of discontinuous Galerkin methods for elliptic problems SIAM J. Numer. Anal. 39 1749 1779 (Pubitemid 35324903)
3. G.A. Baker 1976 Error estimates for finite element methods for second-order hyperbolic equations SIAM J. Numer. Anal. 13 564 576
4. G.A. Baker V.A. Dougalis S.M. Serbin 1979 High order accurate two-step approximations for hyperbolic equations RAIRO. Modél. Math. Anal. Numér. 13 201 226
5. E. Bécache P. Joly C. Tsogka 2000 An analysis of new mixed finite elements for the approximation of wave propagation problems SIAM J. Numer. Anal. 37 1053 1084
6. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
7. Cockburn, B.: Discontinuous Galerkin methods for convection-dominated problems. In: Barth, T., Deconink, H. (eds.) High-Order Methods for Computational Physics. Lect. Notes Comput. Sci. Engrg., vol. 9, pp. 69-224. Springer, Berlin (1999)
8. Cockburn, B., Karniadakis, G.E., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lect. Notes Comput. Sci. Engrg., vol. 11, pp. 3-50. Springer, Berlin (2000)
9. B. Cockburn C.-W. Shu 1989 TVB Runge-Kutta local projection discontinuous Galerkin method for conservation laws II: General framework Math. Comput. 52 411 435
10. B. Cockburn C.-W. Shu 2001 Runge-Kutta discontinuous Galerkin methods for convection-dominated problems J. Sci. Comput. 16 173 261
11. G. Cohen P. Joly J.E. Roberts N. Tordjman 2001 Higher order triangular finite elements with mass lumping for the wave equation SIAM J. Numer. Anal. 38 2047 2078 (Pubitemid 33629645)
12. L.C. Cowsar T.F. Dupont M.F. Wheeler 1990 A-priori estimates for mixed finite element methods for the wave equations Comput. Methods Appl. Mech. Eng. 82 205 222
13. L.C. Cowsar T.F. Dupont M.F. Wheeler 1996 A-priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions SIAM J. Numer. Anal. 33 492 504 (Pubitemid 126431725)
14. Diaz, J., Grote, M.J.: Energy conserving explicit local time-stepping for second-order wave equations. SIAM J. Sci. Comput. (in press)
15. 2-estimates for Galerkin methods for second-order hyperbolic equations SIAM J. Numer. Anal. 10 880 889
16. R.S. Falk G.R. Richter 1999 Explicit finite element methods for symmetric hyperbolic equations SIAM J. Numer. Anal. 36 3 935 952 (Pubitemid 129569418)
17. D.A. French T.E. Peterson 1996 A continuous space-time finite element method for the wave equation Math. Comput. 66 491 506 (Pubitemid 126344319)
18. E. Gekeler 1976 Linear multi-step methods and Galerkin procedures for initial boundary value problems SIAM J. Numer. Anal. 13 536 548
19. T. Geveci 1988 On the application of mixed finite methods to the wave equation RAIRO Modél. Math. Anal. Numér. 22 243 250
20. M.J. Grote A. Schneebeli D. Schötzau 2006 Discontinuous Galerkin finite element method for the wave equation SIAM J. Numer. Anal. 44 2408 2431
21. M.J. Grote A. Schneebeli D. Schötzau 2007 Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates J. Comput. Appl. Math. 204 375 386 (Pubitemid 46617810)
22. 2-norm error estimates IMA J. Numer. Anal. 28 440 468
23. J.S. Hesthaven T. Warburton 2002 Nodal high-order methods on unstructured grids I: Time-domain solution of Maxwell's equations J. Comput. Phys. 181 186 221
24. P. Houston M. Jensen E. Süli 2002 hp-discontinuous Galerkin finite element methods with least-squares stabilization J. Sci. Comput. 17 3 25
25. Hughes, T.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice Hall, New York (1987)
26. P. Monk G.R. Richter 2005 A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media J. Sci. Comput. 22-23 443 477
27. Rivière, B., Wheeler, M.F.: Discontinuous finite element methods for acoustic and elastic wave problems I: Semi-discrete error estimates. Technical Report 01-02, TICAM, UT Austin (2001)
28. Rivière, B., Wheeler, M.F.: Discontinuous finite element methods for acoustic and elastic wave problems. In: ICM2002-Beijing Satellite Conference on Scientific Computing. Contemporary Mathematics, vol. 329, pp. 271-282. Am. Math. Soc., Providence (2003)
29. A. Safjan J.T. Oden 1993 High-order Taylor-Galerkin and adaptive h-p methods for second-order hyperbolic systems: Applications to elastodynamics Comput. Methods Appl. Mech. Eng. 103 187 230 (Pubitemid 23643944)
30. G.R. Shubin J. Bell 1987 A modified equation approach to constructing fourth-order methods for acoustic wave propagation SIAM J. Sci. Comput. 8 135 151