This book presents the notes from the seminar on wave phenomena given in 2019 at the Mathematical Research Center in Oberwolfach.
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.
Author(s): Willy Dörfler, Marlis Hochbruck, Jonas Köhler, Andreas Rieder, Roland Schnaubelt, Christian Wieners
Series: Oberwolfach Seminars, 49
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 367
City: Cham
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
