The convergence of Galerkin approximation schemes for second-order hyperbolic equations with dissipation

Authors:
Barbara Kok and Tunc Geveci

Journal:
Math. Comp. **44** (1985), 379-390, S17

MSC:
Primary 65M10; Secondary 65M60

MathSciNet review:
777270

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Abstract: In this paper we consider certain semidiscrete and fully discrete Galerkin approximations to the solution of an initial-boundary value problem for a second-order hyperbolic equation with a dissipative term. Estimates are obtained in the energy and negative norms associated with the problem, yielding in particular - and -error estimates. The approximation to the initial data is taken, in this case, as the projection with respect to the energy inner product, onto the approximating space. We also obtain estimates for higher-order time derivatives.

**[1]**Garth A. Baker and James H. Bramble,*Semidiscrete and single step fully discrete approximations for second order hyperbolic equations*, RAIRO Anal. Numér.**13**(1979), no. 2, 75–100 (English, with French summary). MR**533876****[2]**Garth A. Baker and Vassilios A. Dougalis,*On the 𝐿^{∞}-convergence of Galerkin approximations for second-order hyperbolic equations*, Math. Comp.**34**(1980), no. 150, 401–424. MR**559193**, 10.1090/S0025-5718-1980-0559193-3**[3]**J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin,*Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations*, SIAM J. Numer. Anal.**14**(1977), no. 2, 218–241. MR**0448926****[4]**James H. Bramble and Vidar Thomée,*Discrete time Galerkin methods for a parabolic boundary value problem*, Ann. Mat. Pura Appl. (4)**101**(1974), 115–152. MR**0388805****[5]**Tunc Geveci,*On the convergence of Galerkin approximation schemes for second-order hyperbolic equations in energy and negative norms*, Math. Comp.**42**(1984), no. 166, 393–415. MR**736443**, 10.1090/S0025-5718-1984-0736443-5**[6]**Reuben Hersh and Tosio Kato,*High-accuracy stable difference schemes for well-posed initial value problems*, SIAM J. Numer. Anal.**16**(1979), no. 4, 670–682. MR**537279**, 10.1137/0716050**[7]**Vidar Thomée,*Negative norm estimates and superconvergence in Galerkin methods for parabolic problems*, Math. Comp.**34**(1980), no. 149, 93–113. MR**551292**, 10.1090/S0025-5718-1980-0551292-5

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DOI:
https://doi.org/10.1090/S0025-5718-1985-0777270-3

Article copyright:
© Copyright 1985
American Mathematical Society