Titlebook: Wave Phenomena; Mathematical Analysi Willy D?rfler,Marlis Hochbruck,Christian Wieners Textbook 2023 The Editor(s) (if applicable) and The A

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Willy D?rfler,Marlis Hochbruck,Jonas K?hler,Andreas Rieder,Roland Schnaubelt,Christian Wieners

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Space-Time Solutions for Linear Hyperbolic SystemsThe linear wave equation can be analyzed in the framework of symmetric Friedrichs systems as a special case of linear hyperbolic conservation laws. Here, we introduce a general framework for the existence and uniqueness of strong and weak solutions in space and time which applies to general linear wave equations.

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Introduction and Local Wellposedness on ,In this section we develop a local wellposedness theory for the quasilinear Maxwell equations on .. Our approach is based on energy methods and a fixed-point argument, which make use of the linear system with time-depending coefficients.

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Local Wellposedness on a DomainIn this chapter we extend the results from the previous one to linear and quasilinear Maxwell systems on a spatial domain ., endowed with boundary conditions. The general theory of symmetric hyperbolic systems is much more sophisticated in this case.

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Exponential Decay Caused by ConductivityIn this chapter we use the wellposedness Theorem . to show global existence and exponential decay to 0 for small initial data in the presence of a strictly positive conductivity ..

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IntroductionSolving wave-type equations numerically requires their discretization either in space and time separately or in space-time. In these lecture notes, we follow a methods-of-lines approach, where we first discretize the problem in space and then in time. For the space discretization, we consider a discontinuous Galerkin finite element method.

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Linear Wave-Type EquationsIn this chapter, we state and analyze the wave-type problem, which we consider within these lecture notes. As mentioned before, we state this problem in a rather general setting, namely in terms of Friedrichs’ operators.

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Space-Time Solutions for Linear Hyperbolic SystemsThe linear wave equation can be analyzed in the framework of symmetric Friedrichs systems as a special case of linear hyperbolic conservation laws. Here, we introduce a general framework for the existence and uniqueness of strong and weak solutions in space and time which applies to general linear wave equations.