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

Authors: Garth A. Baker and Vassilios A. Dougalis
Journal: Math. Comp. 34 (1980), 401-424
MSC: Primary 65M15; Secondary 65N30
MathSciNet review: 559193
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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 ${L^2}$, also possess optimal order rate of convergence in ${L^\infty }$. This is attainable with particular smoothness assumptions on the initial data. We establish sufficient conditions for optimal ${L^\infty }$-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.

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Article copyright: © Copyright 1980 American Mathematical Society