Wavelets in Numerical Simulation

This research monograph addresses recent developments of wavelet concepts in the context of large scale numerical simulation. It offers a systematic attempt to exploit the sophistication of wavelets as a numerical tool by adapting wavelet bases to the problem at hand. This includes both the construction of wavelets on fairly general domains and the adaptation of wavelet bases to the particular structure of function spaces governing certain variational problems. Those key features of wavelets that make them a powerful tool in numerical analysis and simulation are clearly pointed out. The particular constructions are guided by the ultimate goal to ensure the key features also for general domains and problem classes. All constructions are illustrated by figures and examples are given.

Klappentext

Sapere aude! Immanuel Kant (1724-1804) Numerical simulations playa key role in many areas of modern science and technology. They are necessary in particular when experiments for the underlying problem are too dangerous, too expensive or not even possible. The latter situation appears for example when relevant length scales are below the observation level. Moreover, numerical simulations are needed to control complex processes and systems. In all these cases the relevant problems may become highly complex. Hence the following issues are of vital importance for a numerical simulation: - Efficiency of the numerical solvers: Efficient and fast numerical schemes are the basis for a simulation of 'real world' problems. This becomes even more important for realtime problems where the runtime of the numerical simulation has to be of the order of the time span required by the simulated process. Without efficient solution methods the simulation of many problems is not feasible. 'Efficient' means here that the overall cost of the numerical scheme remains proportional to the degrees of freedom, i. e. , the numerical approximation is determined in linear time when the problem size grows e. g. to upgrade accuracy. Of course, as soon as the solution of large systems of equations is involved this requirement is very demanding.

Inhalt

1 Wavelet Bases.- 1.1 Wavelet Bases in L2(?).- 1.1.1 General Setting.- 1.1.2 Characterization of Sobolev-Spaces.- 1.1.3 Riesz Basis Property in L2(?).- 1.1.4 Norm Equivalences.- 1.1.5 General Setting Continued.- 1.1.6 Further Wavelet Features.- 1.1.7 A Program for Constructing Wavelets.- 1.2 Wavelets on the Real Line.- 1.2.1 Orthonormal Wavelets.- 1.2.2 Biorthogonal B-Spline Wavelets.- 1.2.3 Interpolatory Wavelets.- 1.3 Wavelets on the Interval.- 1.3.1 Boundary Scaling Functions.- 1.3.2 Biorthogonal Scaling Functions.- 1.3.3 Biorthogonalization.- 1.3.4 Refinement Matrices.- 1.3.5 Biorthogonal Wavelets on (0, 1).- 1.3.6 Quantitative Aspects of the Biorthogonalization.- 1.3.7 Boundary Conditions.- 1.3.8 Other Bases.- 1.4 Tensor Product Wavelets.- 1.5 Wavelets on General Domains.- 1.5.1 Domain Decomposition and Parametric Mappings.- 1.5.2 Multiresolution and Wavelets on the Subdomains.- 1.5.3 Multiresolution on the Global Domain ?.- 1.5.4 Wavelets on the Global Domain.- 1.5.5 Univariate Matched Wavelets and Other Functions.- 1.5.6 Bivariate Matched Wavelets.- 1.5.7 Trivariate Matched Wavelets.- 1.5.8 Characterization of Sobolev Spaces.- 1.6 Vector Wavelets.- 2 Wavelet Bases for H(div) and H(curl).- 2.1 Differentiation and Integration.- 2.1.1 Differentiation and Integration on the Real Line.- 2.1.2 Differentiation and Integration on (0, 1).- 2.1.3 Assumptions for General Domains.- 2.1.4 Norm Equivalences.- 2.2 The Spaces H(div) and H (curl).- 2.2.1 Stream Function Spaces.- 2.2.2 Flux Spaces.- 2.2.3 Hodge Decompositions.- 2.3 Wavelet Systems for H (curl).- 2.3.1 Wavelets in H0(curl; ?).- 2.3.2 Curl-Free Wavelet Bases.- 2.4 Wavelet Bases for H(div).- 2.4.1 Wavelet Bases in H(div; ?).- 2.4.2 Divergence-Free Wavelet Bases.- 2.5 Helmholtz and Hodge Decompositions.- 2.5.1 A Biorthogonal Helmholtz Decomposition.- 2.5.2 Interrelations and Hodge Decompositions.- 2.6 General Domains.- 2.6.1 Tensor Product Domains.- 2.6.2 Parametric Mappings.- 2.6.3 Fictitious Domain Method.- 2.7 Examples.- 3 Applications.- 3.1 Robust and Optimal Preconditioning.- 3.1.1 Wavelet-Galerkin Discretizations.- 3.1.2 The Lamé Equations for Almost Incompressible Material.- 3.1.3 The Maxwell Equations.- 3.1.4 Preconditioning in H(div; ?).- 3.2 Analysis and Simulation of Turbulent Flows.- 3.2.1 Numerical Simulation of Turbulence.- 3.2.2 Divergence-Free Wavelet Analysis of Turbulence.- 3.2.3 Proper Orthogonal Decomposition (POD).- 3.2.4 Numerical Implementation and Validation.- 3.2.5 Numerical Results I: Data Analysis.- 3.2.6 Numerical Results II: Complexity of Turbulent Flows.- 3.3 Hardening of an Elastoplastic Rod.- 3.3.1 The Physical Problem.- 3.3.2 Numerical Treatment.- 3.3.3 Stress Correction and Wavelet Bases.- 3.3.4 Numerical Results I: Variable Order Discretizations.- 3.3.5 Numerical Results II: Plastic Indicators.- References.- List of Figures.- List of Tables.- List of Symbols.