Wavelets in Numerical Simulation - Problem Adapted Construction and Applications

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 Lame 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.