An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
Authors:
C. Johnson and J. Pitkäranta
Journal:
Math. Comp. 46 (1986), 126
MSC:
Primary 65M60
MathSciNet review:
815828
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Abstract: We prove 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 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. estimates for 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.
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 J. Bergh & J. Löfström, Interpolation Spaces, SpringerVerlag, 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. 112124. MR 0263214 (41:7819)
 [3]
 Ph. Brenner & V. Thomée, "Estimates near discontinuities for some difference schemes," Math. Scand., v. 28, 1971, pp. 329340. 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, SpringerVerlag, Berlin and New York, 1975. MR 0461121 (57:1106)
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 G. Hedström, "The rate of convergence of some difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 363406. 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. 285312. MR 759811 (86a:65103)
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 C. Johnson & J. Pitkäranta, "Convergence of a fully discrete scheme for twodimensional neutron transport," SIAM J. Numer. Anal., v. 20, 1983, pp. 951966. 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 ConvectionDiffusion 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 TwoDimensional, Multigroup, Triangular Mesh, Explicit Neutron Transport Code, LA6735MS, Los Alamos Scientific Laboratory, 1977.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608158284
PII:
S 00255718(1986)08158284
Article copyright:
© Copyright 1986
American Mathematical Society
