A stable finite element method for initialboundary value problems for firstorder hyperbolic systems
Author:
Ragnar Winther
Journal:
Math. Comp. 36 (1981), 6586
MSC:
Primary 65N30
MathSciNet review:
595042
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Abstract: A nonstandard finite element method for initialboundary value problems for firstorder hyperbolic systems in one space dimension with general boundary conditions is analyzed. The method can be considered as a generalization of the box scheme. We first establish a stability result for the method and then derive several error estimates.
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 G. A. Baker, "A finite element method for first order hyperbolic equations," Math. Comp., v. 29, 1975, pp. 9951006. MR 0400744 (53:4574)
 [3]
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 [4]
 J. Douglas, Jr. & T. Dupont, Collocation Methods for Parabolic Equations in a Single Space Variable, SpringerVerlag, New York, 1974. MR 0483559 (58:3551)
 [5]
 T. Dupont, "Galerkin methods for first order hyperbolics: an example," SIAM J. Numer. Anal., v. 10, 1973, pp. 890899. MR 0349046 (50:1540)
 [6]
 M. D. Gunzburger, "On the stability of Galerkin methods for initialboundary value problems for hyperbolic systems," Math. Comp., v. 31, 1977, pp. 661675. MR 0436624 (55:9567)
 [7]
 B. Gustafsson, H.O. Kreiss & A. Sundström, "Stability theory of difference approximations for mixed initial boundary value problems. II," Math. Comp., v. 26, 1972, pp. 649686. MR 0341888 (49:6634)
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 H.O. Kreiss, "Stability theory for difference approximations of mixed initial boundary value problems. I," Math. Comp., v. 22, 1968, pp. 703714. MR 0241010 (39:2355)
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 M. Luskin, "An approximation procedure for nonsymmetric, nonlinear hyperbolic systems with integral boundary conditions," SIAM J. Numer. Anal., v. 16, 1979, pp. 145164. MR 518690 (80c:65214)
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 V. Thomée, "A stable difference scheme for the mixed boundary value problem for a hyperbolic, firstorder system in two dimensions," J. Soc. Indust. Appl. Math., v. 10, 1962, pp. 229245. MR 0154422 (27:4370)
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 V. Thomée, "Estimates of the FriedrichsLewy type for mixed problems in the theory of linear hyperbolic differential equations in two independent variables," Math. Scand., v. 5, 1957, pp. 93113. MR 0092082 (19:1059d)
 [12]
 V. Thomée, "Negative norm estimates and superconvergence in Galerkin methods for parabolic problems," Math. Comp., v. 34, 1980, pp. 93113. MR 551292 (81a:65092)
 [13]
 R. Winther, "A conservative finite element method for the Kortewegde Vries equations," Math. Comp., v. 34, 1980, pp. 2343. MR 551289 (81a:65108)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198105950426
PII:
S 00255718(1981)05950426
Article copyright:
© Copyright 1981
American Mathematical Society
