On the L°°-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 2 °° order rate of convergence in L , 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 L -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.