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

Abstract

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 L 2 {L^2} , also possess optimal order rate of convergence in L ∞ {L^\infty } . This is attainable with particular smoothness assumptions on the initial data. We establish sufficient conditions for optimal L ∞ {L^\infty } -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.