Abstract
Efficient procedures for time-stepping Galerkin methods for approximating smooth solutions of quasilinear second-order hyperbolic equations are considered. The techniques presented can be used to analyze approximation procedures for related second-order-in-time quasilinear partial differential equations which have applications including initial-boundary value problems for vibrations (possibly) with inertia, dynamics of rotating fluids, and nonlinear viscoelasticity. The procedure involves the use of a pre-conditioned iterative method for approximately solving the different linear systems of equations arising at each time step in a discrete-time Galerkin method. Optimal order L 2 spatial errors and almost optimal order work estimates are obtained for the second-order hyperbolic case.
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