Wavelets in Numerical Simulation

by Urban, Karsten

• ISBN13: 9783540430551
• ISBN10: 3540430555
• Format: Paperback
• Copyright: 2002-07-01
• Publisher: Springer Nature

Summary

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.

Table of Contents

Preface	vii
Wavelet Bases	1	(82)
Wavelet Bases in L2(Ω)	1	(11)
General Setting	2	(3)
Characterization of Soboley-Spaces	5	(2)
Riesz Basis Property in L2(Ω)	7	(1)
Norm Equivalences	8	(1)
General Setting Continued	9	(1)
Further Wavelet Features	10	(1)
A Program for Constructing Wavelets	11	(1)
Wavelets on the Real Line	12	(5)
Orthonormal Wavelets	13	(1)
Biorthogonal B-Spline Wavelets	14	(1)
Interpolatory Wavelets	15	(2)
Wavelets on the Interval	17	(29)
Boundary Scaling Functions	19	(1)
Biorthogonal Scaling Functions	19	(6)
Biorthogonalization	25	(5)
Refinement Matrices	30	(3)
Biorthogonal Wavelets on (0, 1)	33	(8)
Quantitative Aspects of the Biorthogonalization	41	(3)
Boundary Conditions	44	(1)
Other Bases	45	(1)
Tensor Product Wavelets	46	(1)
Wavelets on General Domains	47	(34)
Domain Decomposition and Parametric Mappings	51	(2)
Multiresolution and Wavelets on the Subdomains	53	(1)
Multiresolution on the Global Domain Ω	54	(1)
Wavelets on the Global Domain	55	(1)
Univariate Matched Wavelets and Other Functions	56	(6)
Bivariate Matched Wavelets	62	(11)
Trivariate Matched Wavelets	73	(5)
Characterization of Sobolev Spaces	78	(3)
Vector Wavelets	81	(2)
Wavelet Bases for H(div) and H(curl)	83	(26)
Differentiation and Integration	84	(6)
Differentiation and Integration on the Real Line	84	(1)
Differentiation and Integration on (0, 1)	85	(1)
Assumptions for General Domains	86	(3)
Norm Equivalences	89	(1)
The Spaces H(div) and H (curl)	90	(2)
Stream Function Spaces	91	(1)
Flux Spaces	91	(1)
Hodge Decompositions	92	(1)
Wavelet Systems for H (curl)	92	(6)
Wavelets in H0(curl; Ω)	93	(2)
Curl-Free Wavelet Bases	95	(3)
Wavelet Bases for H (div)	98	(2)
Wavelet Bases in H (div; Ω)	98	(1)
Divergence-Free Wavelet Bases	99	(1)
Helmholtz and Hodge Decompositions	100	(2)
A Biorthogonal Helmholtz Decomposition	100	(1)
Interrelations and Hodge Decompositions	101	(1)
General Domains	102	(3)
Tensor Product Domains	102	(1)
Parametric Mappings	103	(1)
Fictitions Domain Method	104	(1)
Examples	105	(4)
Applications	109	(52)
Robust and Optimal Preconditioning	109	(9)
Wavelet-Galerkin Discretizations	109	(3)
The Lame Equations for Almost Incompressible Material	112	(4)
The Maxwell Equations	116	(1)
Preconditioning in H (div; Ω)	117	(1)
Analysis and Simulation of Turbulent Flows	118	(20)
Numerical Simulation of Turbulence	118	(1)
Divergence-Free Wavelet Analysis of Turbulence	119	(1)
Proper Orthogonal Decomposition (POD)	120	(2)
Numerical Implementation and Validation	122	(2)
Numerical Results I: Data Analysis	124	(4)
Numerical Results II: Complexity of Turbulent Flows	128	(10)
Hardening of an Elastoplastic Rod	138	(23)
The Physical Problem	138	(4)
Numerical Treatment	142	(3)
Stress Correction and Wavelet Bases	145	(1)
Numerical Results I: Variable Order Discretizations	146	(8)
Numerical Results II: Plastic Indicators	154	(7)
References	161	(8)
List of Figures	169	(4)
List of Tables	173	(2)
List of Symbols	175	(4)
Index	179