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On the convergence of Galerkin approximation schemes for second-order hyperbolic equations in energy and negative norms


Author: Tunc Geveci
Journal: Math. Comp. 42 (1984), 393-415
MSC: Primary 65M60
DOI: https://doi.org/10.1090/S0025-5718-1984-0736443-5
MathSciNet review: 736443
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Abstract: Given certain semidiscrete and single step fully discrete Galerkin approximations to the solution of an initial-boundary value problem for a second-order hyperbolic equation, $ {H^1}$ and $ {L^2}$ error estimates are obtained. These estimates are valid simultaneously when the approximation to the initial data is taken to be the projection onto the approximating space with respect to the inner product which induces the energy norm that is naturally associated with the problem. The $ {L^2}$-estimate is obtained as a by-product of the analysis of convergence in certain negative norms. Estimates are also obtained for the convergence of higher-order time derivatives in the presence of sufficiently smooth data.


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  • [1] G. A. Baker & J. H. Bramble, "Semidiscrete and single step fully discrete approximations for second order hyperbolic equations," RAIRO Anal. Numér., v. 13, 1979, pp. 75-100. MR 533876 (80f:65115)
  • [2] G. A. Baker, J. H. Bramble & V. Thomée, "Single step Galerkin approximations for parabolic problems," Math. Comp., v. 31, 1977, pp. 818-847. MR 0448947 (56:7252)
  • [3] G. A. Baker & V. A. Dougalis, "On the $ {L^\infty }$-convergence of Galerkin approximations for second-order hyperbolic equations," Math. Comp., v. 34, 1980, pp. 401-424. MR 559193 (81f:65066)
  • [4] J. H. Bramble & A. H. Schatz, "Higher order local accuracy by averaging in the finite element method," Math. Comp., v. 31, 1977, pp. 94-111. MR 0431744 (55:4739)
  • [5] J. H. Bramble, A. H. Schatz, V. Thomée & L. B. Wahlbin, "Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations," SIAM J. Numer. Anal., v. 14, 1977, pp. 218-241. MR 0448926 (56:7231)
  • [6] J. H. Bramble & V. Thomée, "Discrete time Galerkin methods for a parabolic boundary value problem," Ann. Mat. Pura Appl. (4), v. 101, 1974, pp. 115-152. MR 0388805 (52:9639)
  • [7] J. E. Marsden & T. J. R. Hughes, "Topics in the mathematical foundations of elasticity," in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium vol. II (R. J. Knops (Ed.)), Pitman, London, San Francisco, Melbourne, 1978. MR 576231 (81e:73003)
  • [8] G. Strang & G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, N. J., 1973. MR 0443377 (56:1747)
  • [9] V. Thomée, "Negative norm estimates and superconvergence in Galerkin methods for parabolic problems," Math. Comp., v. 34, 1980, pp. 93-113. MR 551292 (81a:65092)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1984-0736443-5
Article copyright: © Copyright 1984 American Mathematical Society

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