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Full Description
This book provides good coverage of the powerful numerical techniques namely, finite element and wavelets, for the solution of partial differential equation to the scientists and engineers with a modest mathematical background. The objective of the book is to provide the necessary mathematical foundation for the advanced level applications of these numerical techniques. The book begins with the description of the steps involved in finite element and wavelets-Galerkin methods. The knowledge of Hilbert and Sobolev spaces is needed to understand the theory of finite element and wavelet-based methods. Therefore, an overview of essential content such as vector spaces, norm, inner product, linear operators, spectral theory, dual space, and distribution theory, etc. with relevant theorems are presented in a coherent and accessible manner. For the graduate students and researchers with diverse educational background, the authors have focused on the applications of numerical techniques which are developed in the last few decades. This includes the wavelet-Galerkin method, lifting scheme, and error estimation technique, etc.
Features:
• Computer programs in Mathematica/Matlab are incorporated for easy understanding of wavelets.
• Presents a range of workout examples for better comprehension of spaces and operators.
• Algorithms are presented to facilitate computer programming.
• Contains the error estimation techniques necessary for adaptive finite element method.
This book is structured to transform in step by step manner the students without any knowledge of finite element, wavelet and functional analysis to the students of strong theoretical understanding who will be ready to take many challenging research problems in this area.
Contents
Preface
Authors
1. Overview of finite element method
Some common governing differential equations
Basic steps of finite element method
Element stiffness matrix for a bar
Element stiffness matrix for single variable 2d element
Element stiffness matrix for a beam element
References for further reading
2. Wavelets
Wavelet basis functions
Wavelet-Galerkin method
Daubechies wavelets for boundary and initial value problems
References for further reading
3. Fundamentals of vector spaces
Introduction
Vector spaces
Normed linear spaces
Inner product spaces
Banach spaces
Hilbert spaces
Projection on finite dimensional spaces
Change of basis - Gram-Schmidt othogonalization process
Riesz bases and frame conditions
References for further reading
4. Operators
General concept of functions
Operators
Linear and adjoint operators
Functionals and dual space
Spectrum of bounded linear self-adjoint operator
Classification of differential operators
Existence, uniqueness and regularity of solution
References
5. Theoretical foundations of the finite element method
Distribution theory
Sobolev spaces
Variational Method
Nonconforming elements and patch test
References for further reading
6. Wavelet- based methods for differential equations
Fundamentals of continuous and discrete wavelets
Multiscaling
Classification of wavelet basis functions
Discrete wavelet transform
Lifting scheme for discrete wavelet transform
Lifting scheme to customize wavelets
Non-standard form of matrix and its solution
Multigrid method
References for further reading
7. Error - estimation
Introduction
A-priori error estimation
Recovery based error estimators
Residual based error estimators
Goal oriented error estimators
Hierarchical and wavelet based error estimator
References for further reading
Appendices
