A finite element method for firstorder hyperbolic systems
Author:
Mitchell Luskin
Journal:
Math. Comp. 35 (1980), 10931112
MSC:
Primary 65N30; Secondary 65M15, 76N15
MathSciNet review:
583489
Fulltext PDF Free Access
Abstract 
References 
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Abstract: A new finite element method is proposed for the numerical solution of a class of initialboundary value problems for firstorder hyperbolic systems in one space dimension. An application of our procedure to a system modeling gas flow in a pipe is discussed. Asymptotic error estimates are derived in the norm in space.
 [1]
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T. DUPONT & L. WAHLBIN, " optimality of weighted projections into piecewise polynomial spaces," Manuscript, Dept. of Math., Univ. of Chicago, 1974.
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G.
W. Hedstrom, The Galerkin method based on Hermite cubics, SIAM
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(80i:65116), http://dx.doi.org/10.1137/0716032
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 [7]
Mitchell
Luskin, On the existence of global smooth solutions for a model
equation for fluid flow in a pipe, J. Math. Anal. Appl.
84 (1981), no. 2, 614–630. MR 639688
(83g:76078), http://dx.doi.org/10.1016/0022247X(81)90192X
 [8]
M. LUSKIN, "A finite element method for first order hyperbolic systems in two space variables," Manuscript, Dept. of Math., Univ. of Michigan, 1978.
 [9]
G. PLATZMAN, "Normal modes of the world ocean. Part 1. Design of a finiteelement barotropic model," J. Phys. Oceanogr., v. 8, 1979, pp. 323343.
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H. Rachford Jr., Twolevel discretetime Galerkin approximations
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SIAM J. Numer. Anal. 10 (1973), 1010–1026. MR 0339519
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PrenticeHall Inc., Englewood Cliffs, N. J., 1973. PrenticeHall Series in
Automatic Computation. MR 0443377
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 [12]
V. STREETER, Fluid Mechanics, 5th ed., McGrawHill, New York, 1971.
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Lars
B. Wahlbin, A dissipative Galerkin method applied to some
quasilinear hyperbolic equations, Rev. Française Automat.
Informat. Recherche Opérationnelle Sér. Rouge
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 [1]
 J. DOUGLAS, JR., T. DUPONT & L. WAHLBIN, "Optimal error estimates for Galerkin approximations to solutions of twopoint boundary value problems," Math. Comp., v. 29, 1975, pp. 475483. MR 0371077 (51:7298)
 [2]
 T. DUPONT, "Galerkin methods for modeling gas pipelines," Constructive and Computational Methods for Differential and Integral Equations, Lecture Notes in Math., Vol. 430, SpringerVerlag, Berlin and New York, 1974. MR 0502035 (58:19223)
 [3]
 T. DUPONT, "Galerkin methods for first order hyperbolics: an example," SIAM J. Numer. Anal., v. 10, 1973, pp. 890899. MR 0349046 (50:1540)
 [4]
 T. DUPONT & L. WAHLBIN, " optimality of weighted projections into piecewise polynomial spaces," Manuscript, Dept. of Math., Univ. of Chicago, 1974.
 [5]
 G. HEDSTROM, "The Galerkin method based on Hermite cubics," SIAM J. Numer. Anal., v. 16, 1979, pp. 385393. MR 530476 (80i:65116)
 [6]
 M. LEES, "A linear three level difference scheme for quasilinear parabolic equations," Math. Comp., v. 20, 1966, pp. 516522. MR 0207224 (34:7040)
 [7]
 M. LUSKIN, "On the existence of global smooth solutions for a model equation for fluid flow in a pipe," Manuscript, Dept. of Math., Ecole Polytechnique Fédérale de Lausanne, 1980. MR 639688 (83g:76078)
 [8]
 M. LUSKIN, "A finite element method for first order hyperbolic systems in two space variables," Manuscript, Dept. of Math., Univ. of Michigan, 1978.
 [9]
 G. PLATZMAN, "Normal modes of the world ocean. Part 1. Design of a finiteelement barotropic model," J. Phys. Oceanogr., v. 8, 1979, pp. 323343.
 [10]
 H. RACHFORD, JR., "Twolevel discretetime Galerkin approximations for second order nonlinear parabolic partial differential equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 10101026. MR 0339519 (49:4277)
 [11]
 G. STRANG & G. FIX, An Analysis of the Finite Element Method, PrenticeHall, Englewood Cliffs, N.J., 1973. MR 0443377 (56:1747)
 [12]
 V. STREETER, Fluid Mechanics, 5th ed., McGrawHill, New York, 1971.
 [13]
 L. WAHLBIN, "A dissipative Galerkin method applied to some quasilinear hyperbolic equations," R.A.I.R.O. Anal. Numer., v. 8, 1974, pp. 109117. MR 0368447 (51:4688)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005834892
PII:
S 00255718(1980)05834892
Article copyright:
© Copyright 1980 American Mathematical Society
