Wave Phenomena: Mathematical Analysis and Numerical Approximation (Oberwolfach Seminars)

Preface
Acknowledgements
Contents
About the Authors
Part I Space-Time Approximations for Linear Acoustic, Elastic, and Electro-Magnetic Wave Equations
Willy Dörfler and Christian Wieners
	1 Modeling of Acoustic, Elastic, and Electro-Magnetic Waves
		1.1 Modeling in Continuum Mechanics
		1.2 The Wave Equation in 1d
		1.3 Harmonic, Anharmonic and Viscous Waves
		1.4 Elastic Waves
		1.5 Visco-Elastic Waves
		1.6 Acoustic Waves in Solids
		1.7 Electro-Magnetic Waves
	2 Space-Time Solutions for Linear Hyperbolic Systems
		2.1 Linear Hyperbolic First-Order Systems
		2.2 Solution Spaces
		2.3 Solution Concepts
		2.4 Existence and Uniqueness of Space-Time Solutions
		2.5 Mapping Properties of the Space-Time Operator
		2.6 Inf-Sup Stability
		2.7 Applications to Acoustics and Visco-Elasticity
	3 Discontinuous Galerkin Methods for Linear Hyperbolic Systems
		3.1 Traveling Wave Solutions in Homogeneous Media
		3.2 Reflection of Traveling Acoustic Waves at Boundaries
		3.3 Transmission and Reflection of Traveling Waves at Interfaces
		3.4 The Riemann Problem for Acoustic Waves
		3.5 The Riemann Problem for Linear Conservation Laws
		3.6 The DG Discretization with Full Upwind
		3.7 The Full Upwind Discretization for the Wave Equation
	4 A Petrov–Galerkin Space-Time Approximation for Linear Hyperbolic Systems
		4.1 Decomposition of the Space-Time Cylinder
		4.2 The Petrov–Galerkin Setting
		4.3 Inf-Sup Stability
		4.4 Convergence for Strong Solutions
		4.5 Convergence for Weak Solutions
		4.6 Goal-Oriented Adaptivity
		4.7 Reliable Error Estimation for Weak Solutions
	Part II Local Wellposedness and Long-Time Behavior of Quasilinear Maxwell Equations
Roland Schnaubelt
	5 Introduction and Local Wellposedness on R3
		5.1 The Maxwell System
		5.2 The Linear Problem on R3 in L2
		5.3 The Linear Problem on R3 in H3
		5.4 The Quasilinear Problem on R3
		5.5 Energy and Blow-Up
	6 Local Wellposedness on a Domain
		6.1 The Maxwell System on a Domain
		6.2 The Linear Problem on R3+ in L2
		6.3 The Linear Problem on R3+ in H3
		6.4 The Quasilinear Problem on R+3
		6.5 The Main Wellposedness Result
	7 Exponential Decay Caused by Conductivity
		7.1 Introduction and Theorem on Decay
		7.2 Energy and Observability-Type Inequalities
		7.3 Time Regularity Controls Space Regularity
	Part III Error Analysis of Second-Order Time Integration Methods for Discontinuous Galerkin Discretizations of Friedrichs\' Systems
Marlis Hochbruck and Jonas Köhler
	8 Introduction
		Acknowledgment
		8.1 Notation
	9 Linear Wave-Type Equations
		9.1 A Short Course on Semigroup Theory
		9.2 Analytical Setting and Friedrichs\' Operators
		9.3 Examples
			9.3.1 Advection Equation
			9.3.2 Acoustic Wave Equation
			9.3.3 Maxwell Equations
	10 Spatial Discretization
		10.1 The Discrete Setting
		10.2 Friedrichs\' Operators in the Discrete Setting
		10.3 Discrete Friedrichs\' Operators
		10.4 The Spatially Semidiscrete Problem
	11 Full Discretization
		11.1 Crank–Nicolson Scheme
		11.2 Leapfrog Scheme
		11.3 Peaceman–Rachford Scheme
		11.4 Locally Implicit Scheme
		11.5 Addendum
	12 Error Analysis
		12.1 Crank–Nicolson Scheme
			12.1.1 Error Recursion
			12.1.2 Bounds on the Defect
			12.1.3 Error Bounds for the dG-Crank–Nicolson Scheme
		12.2 Leapfrog Scheme
			12.2.1 Error Recursion
			12.2.2 Bounds on and Splitting of the Defect
			12.2.3 Error Bounds for the dG-Leapfrog Scheme
		12.3 Peaceman–Rachford Scheme
			12.3.1 Error Recursion
			12.3.2 Bounds on the Defect
			12.3.3 Error Bounds for the dG-Peaceman–Rachford Scheme
		12.4 Locally Implicit Scheme
			12.4.1 Error Recursion
			12.4.2 Bounds on and Splitting of the Defect
			12.4.3 Error Bounds for the dG-Locally Implicit Scheme
		12.5 Concluding Remarks
			12.5.1 Less Regular Solutions
			12.5.2 Approximations of Initial Values
			12.5.3 Approximations at Half Time Steps
	13 Appendix
		13.1 Friedrichs\' Operators Exhibiting a Two-Field Structure
		13.2 Full Bounds for the Discrete Derivative Errors
	14 List of Definitions
	Part IV An Abstract Framework for Inverse Wave Problems with Applications
Andreas Rieder
	15 What Is an Inverse and Ill-Posed Problem?
		15.1 Electric Impedance Tomography: The Continuum Model
		15.2 Seismic Tomography
	16 Local Ill-Posedness
		16.1 Examples for Local Ill-Posedness
			16.1.1 Electric Impedance Tomography
			16.1.2 Seismic Tomography
		16.2 Linearization and Ill-Posedness
	17 Regularization of Linear Ill-Posed Problems in Hilbert Spaces
	18 Newton-Like Solvers for Non-linear Ill-Posed Problems
		18.1 Decreasing Error and Weak Convergence
			18.1.1 A Heuristic for Choosing the Tolerances
		18.2 Convergence Without Noise
		18.3 Regularization Property of REGINN
	19 Inverse Problems Related to Abstract Evolution Equations
		19.1 Motivation: Full Waveform Inversion in Seismic Imaging
			19.1.1 Elastic Wave Equation
			19.1.2 Visco-Elastic Wave Equation
			19.1.3 The Inverse Problem of Seismic Imaging in the Visco-Elastic Regime
			19.1.4 Visco-Elastic Wave Equation (Transformed)
		19.2 Abstract Framework
			19.2.1 Existence, Uniqueness, and Regularity
			19.2.2 Parameter-to-Solution Map
	20 Applications
		20.1 Full Waveform Inversion in the Visco-Elastic Regime
			20.1.1 Full Waveform Forward Operator
			20.1.2 Differentiability and Adjoint
		20.2 Maxwell\'s Equation: Inverse Electromagnetic Scattering
			20.2.1 Inverse Electromagnetic Scattering
			20.2.2 The Electromagnetic Forward Map
			20.2.3 Differentiability and Adjoint
	References