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The research on wave-type problems is a fascinating and emerging field in mathematical research with many challenging applications in sciences and engineering. Profound investigations on waves require a strong interaction of several mathematical disciplines including functional analysis, partial differential equations, mathematical modeling, mathematical physics, numerical analysis, and scientific computing.
The goal of this book is to present a comprehensive introduction to the research on wave phenomena. Starting with basic models for acoustic, elastic, and electro-magnetic waves, topics such as the existence of solutions for linear and some nonlinear material laws, efficient discretizations and solution methods in space and time, and the application to inverse parameter identification problems are covered. The aim of this book is to intertwine analysis and numerical mathematics for wave-type problems promoting thus cooperative research projects in this field.
Space-Time Approximations for Linear Acoustic, Elastic, and Electro-Magnetic Wave Equations
Modeling of Acoustic, Elastic, and Electro-Magnetic Waves
Space-Time Solutions for Linear Hyperbolic Systems
Discontinuous Galerkin Methods for Linear Hyperbolic Systems
A Petrov–Galerkin Space-Time Approximation for Linear Hyperbolic Systems
Local Wellposedness and Long-Time Behavior of Quasilinear Maxwell Equations
Introduction and Local Wellposedness on R3
Local Wellposedness on a Domain
Exponential Decay Caused by Conductivity
Error Analysis of Second-Order Time Integration Methods for Discontinuous Galerkin Discretizations of Friedrichs’ Systems
Introduction
Linear Wave-Type Equations
Spatial Discretization
Full Discretization
Error Analysis
Appendix
List of Definitions
An Abstract Framework for Inverse Wave Problems with Applications
What Is an Inverse and Ill-Posed Problem?
Local Ill-Posedness
Regularization of Linear Ill-Posed Problems in Hilbert Spaces
Newton-Like Solvers for Non-linear Ill-Posed Problems
Inverse Problems Related to Abstract Evolution Equations
Applications