On the ^{∞}-convergence of Galerkin approximations for second-order hyperbolic equations

It is shown that certain classes of high order accurate Galerkin approximations for homogeneous second-order hyperbolic equations, known to possess optimal order rate of convergence in L2, also possess optimal order rate of convergence in L This is attainable with particular smoothness assumptions on the initial data. We establish sufficient conditions for optimal L00-convergence of the approximations to the solution and also the approximation to its time derivative. This is done for both semidiscrete approximations and for single-step fully discrete approximations generated by rational functions.

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