Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation

Authors: C. Johnson and J. Pitkäranta
Journal: Math. Comp. 46 (1986), 1-26
MSC: Primary 65M60
MathSciNet review: 815828
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove $ {L_p}$ stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis techniques, we obtain $ {L_2}$ estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. $ {L_p}$ estimates for $ p \ne 2$ are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree. The latter estimates are proved by combining finite difference and finite element analysis techniques.

References [Enhancements On Off] (What's this?)

  • [1] J. Bergh & J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin and New York, 1976.
  • [2] J. H. Bramble & S. R. Hilbert, "Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation," SIAM J. Numer. Anal., v. 7, 1970, pp. 112-124. MR 0263214 (41:7819)
  • [3] Ph. Brenner & V. Thomée, "Estimates near discontinuities for some difference schemes," Math. Scand., v. 28, 1971, pp. 329-340. MR 0305613 (46:4743)
  • [4] Ph. Brenner, V. Thomée & L. Wahlbin, Besov Spaces and Applications to Difference Methods for Initial Value Problems, Lecture Notes in Math., Vol. 434, Springer-Verlag, Berlin and New York, 1975. MR 0461121 (57:1106)
  • [5] G. Hedström, "The rate of convergence of some difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 363-406. MR 0230489 (37:6051)
  • [6] C Johnson, U. Nävert & J. Pitkäranta, "Finite element methods for linear hyperbolic problems," Comput. Methods Appl. Mech. Engrg., v. 45, 1984, pp. 285-312. MR 759811 (86a:65103)
  • [7] C. Johnson & J. Pitkäranta, "Convergence of a fully discrete scheme for two-dimensional neutron transport," SIAM J. Numer. Anal., v. 20, 1983, pp. 951-966. MR 714690 (85c:82082)
  • [8] C. Johnson & Mingyoung Huang, "An analysis of the discontinuous Galerkin method for Friedrichs systems." (To appear.)
  • [9] P. Lesaint & P.-A. Raviart, "On a finite element method for solving the neutron transport equation," in Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed.), Academic Press, New York, 1974. MR 0658142 (58:31918)
  • [10] U. Nävert, A Finite Element Method for Convection-Diffusion Problems, Thesis, Department of Computer Science, Chalmers University of Technology, 1982.
  • [11] W. H. Reed, T. R. Hill, F. W. Brinkley & K. D. Lathrop, TRIDENT, a Two-Dimensional, Multigroup, Triangular Mesh, Explicit Neutron Transport Code, LA-6735-MS, Los Alamos Scientific Laboratory, 1977.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M60

Retrieve articles in all journals with MSC: 65M60

Additional Information

Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society