On the convergence of Galerkin approximation schemes for second-order hyperbolic equations in energy and negative norms. (English) Zbl 0553.65082

H\({}^ 1\) and \(L^ 2\) error estimates are obtained for certain semidiscrete and single step fully discrete Galerkin approximations to the solution of an initial boundary value problem. 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.
Reviewer: W.Ames


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
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