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

Baker, Garth A.; Dougalis, Vassilios A.

doi:10.1090/S0025-5718-1980-0559193-3

On the $L\sp{\infty }$ -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
DOI: https://doi.org/10.1090/S0025-5718-1980-0559193-3
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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