Publication Cover
Applicable Analysis
An International Journal
Volume 100, 2021 - Issue 6
180
Views
2
CrossRef citations to date
0
Altmetric
Articles

Unconditional stability and optimal error estimates of discontinuous Galerkin methods for the second-order wave equation

, , &
Pages 1143-1157 | Received 23 Jan 2019, Accepted 24 Jun 2019, Published online: 08 Jul 2019

References

  • Arnold DN. An interior penalty finite element method with discontinuous elements. SIAM J Numer Anal. 1982;19:742–760. doi: 10.1137/0719052
  • Douglas J, Dupont T. Interior penalty procedures for elliptic and parabolic Galerkin methods. Berlin: Springer-Verlag; 1976. (Lecture notes in phys.; 58).
  • Cockburn B, Kanschat G, Schötzau D. A locally conservative LDG method for the incompressible Navier-Stokes equations. Math Comp. 2005;74:1067–1095. doi: 10.1090/S0025-5718-04-01718-1
  • Cockburn B, Shu C-W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J Numer Anal. 1998;35:2440–2463. doi: 10.1137/S0036142997316712
  • Perugia I, Schötzau D. An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J Sci Comput. 2002;17:561–571. doi: 10.1023/A:1015118613130
  • Cockburn B, Gepalakrishnan J, Lazarov R. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J Numer Anal. 2009;47:1319–1365. doi: 10.1137/070706616
  • Castillo P, Cockburn B, Schötzau D, et al. Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math Comp. 2002;71:455–479. doi: 10.1090/S0025-5718-01-01317-5
  • Bey K, Oden J. hp-version discontinuous Galerkin methods for hyperbolic conservation laws. Comput Methods Appl Mech Eng. 1996;133:259–286. doi: 10.1016/0045-7825(95)00944-2
  • Grote M, Schneebeli A, Schötzau D. Discontinuous Galerkin finite element method for the wave equation. SIAM J Numer Anal. 2006;44:2408–2431. doi: 10.1137/05063194X
  • Grote M, Schötzau D. Optimal error estimates for the fully discrete interior penalty DG method for the wave equation. J Sci Comput. 2009;40:257–272. doi: 10.1007/s10915-008-9247-z
  • Houston P, Schwab C, Süli E. Stabilized hp-finite element methods for hyperbolic problems. SIAM J Numer Anal. 2000;37:1618–1643. doi: 10.1137/S0036142998348777
  • Bassi F, Rebay S. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J Comput Phys. 1997;131:267–279. doi: 10.1006/jcph.1996.5572
  • Hu C, Shu C-W. A discontinuous Galerkin finite element method for Hamilton-Jacobi equations. SIAM J Sci Comput. 1999;21:666–690. doi: 10.1137/S1064827598337282
  • Kornhuber R, Lepsky O, Hu C, et al. The analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations. Appl Numer Math. 2000;33:423–434. doi: 10.1016/S0168-9274(99)00109-9
  • Han W, Huang J, Eichholz J. Discrete-ordinate discontinuous Galerkin methods for solving the radiative transfer equation. SIAM J Sci Comput. 2010;32:477–497. doi: 10.1137/090767340
  • Wang F, Han W, Cheng X. Discontinuous Galerkin methods for solving elliptic variational inequalities. SIAM J Numer Anal. 2010;48:708–733. doi: 10.1137/09075891X
  • Wang F, Han W, Cheng X. Discontinuous Galerkin methods for solving Signorini problem. IMA J Numer Anal. 2011;31:1754–1772. doi: 10.1093/imanum/drr010
  • Wang F, Han W, Cheng X. Discontinuous Galerkin methods for solving a quasistatic contact problem. Numer Math. 2014;126:771–800. doi: 10.1007/s00211-013-0574-0
  • Wang F, Han W, Eichholz J, et al. A posteriori error estimates of discontinuous Galerkin methods for obstacle problems. Nonlinear Anal: Real World Appl. 2015;22:664–679. doi: 10.1016/j.nonrwa.2014.08.011
  • Wang F, Zhang T, Han W. C0 discontinuous Galerkin methods for a Kirchhoff plate contact problem. J Comput Math. 2019;37:184–200. doi: 10.4208/jcm.1711-m2017-0187
  • Han W, He L, Wang F. Optimal order error estimates for discontinuous Galerkin methods for the wave equation. J Sci Comput. 2019;78:121–144. doi: 10.1007/s10915-018-0755-1
  • Lions J-L, Magenes E. Non-Homogeneous boundary value problems and applications. Vol. I. New York (NY): Springer-Verlag; 1972.
  • Evans LC. Partial differential equations, graduate studies in mathematics. Vol. 19. Providence (RI): American Mathematical Society; 1998.
  • Riviere B. Discontinuous Galerkin methods for solving elliptic and parabolic equations, Theory and implementation. Philadelphia (PA): Society for Industrial and Applied Mathematics; 2008.
  • Arnold DN, Brezzi F, Cockburn B, Marini LD. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal. 2002;39:1749–1779. doi: 10.1137/S0036142901384162
  • Bassi F, Rebay S, Mariotti G, et al. A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In: Decuypere R, Dibelius G, editors. Proceedings of 2nd European conference on turbomachinery, fluid dynamics and thermodynamics. Antwerpen: Technologisch Instituut; 1997. p. 99–108.
  • Brezzi F, Manzini G, Marini D, et al. Discontinuous finite elements for diffusion problems, in Atti Convegno in onore di F. Brioschi (Milan, 1997). Milan: Istituto Lombardo, Accademia di Scienze e Lettere; 1999. p. 197–217.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.