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Description:
This thesis provides a unified framework for the error analysis for space and time discretizations of a quite general class of quasilinear wave-type problems. For the space discretization we prove a rigorous error estimate based on semigroup theory for nonautonomous problems. Compared to previous results, which are mostly based on Banach’s fixed-point theorem, this approach allows for a better insight into the individual error contributions. Furthermore, since wellposedness results for quasilinear wave-type problems are in general based on severe regularity assumptions with respect to the boundary of the domain, we consider nonconforming space discretizations in order to allow for domain approximation. Furthermore, we provide a rigorous error analysis for the full discretization with three different one-step time-integration schemes. On the one hand, we consider the implicit midpoint rule and a linearized version thereof. On the other hand, we also investigate the leapfrog scheme, which is an explicit scheme. Throughout this thesis, we illustrate the relevance of the abstract framework by application of our results to the undamped Westervelt equation and the Maxwell equations with Kerr nonlinearity. Finally, we conclude with numerical examples.