Wave Phenomena: Mathematical Analysis and Numerical Approximation

Description

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. 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