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Mathematical theory of subdivision : finite element and wavelet methods /

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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 found...

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Bibliographic Details
Main Authors: Kumar, Sandeep (Professor of mechanical engineering) (Author), Pathak, Ashish (Author), Khan, Debasish (Author)
Corporate Author: Taylor & Francis
Format: eBook
Language:English
Published: Boca Raton, Florida : CRC Press, [2019]
Subjects:
Finite element method.
Wavelets (Mathematics)
Numerical analysis.
Harmonic analysis.
Generalized spaces.
MATHEMATICS / General
MATHEMATICS / Applied
MATHEMATICS / Number Systems
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • <P>Preface</P><P>About the authors</P><OL><B><P></P></OL><P>1. Overview of finite element method</P><OL><P></P><OL></B><P><LI>Some common governing differential equations </LI><P></P><P><LI>Basic steps of finite element method </LI><P></P><P><LI>Element stiffness matrix for a bar </LI><P></P><P><LI>Element stiffness matrix for single variable 2d element </LI><P></P><P><LI>Element stiffness matrix for a beam element</LI><P></P><P><LI>References for further reading</LI><P></P></OL></OL><OL><B><P></P></OL><P>2. Wavelets</P><OL><P></P></OL><OL><OL></B><P><LI>Wavelet basis functions</LI><P></P><P><LI>Wavelet-Galerkin method </LI><P></P><P><LI>Daubechies wavelets for boundary and initial value problems </LI><P></P><P><LI>References for further reading</LI><P></P></OL></OL><OL><B><P></P></OL><P>3. Fundamentals of vector spaces </P><OL><P></P></OL><OL><OL></B><P><LI>Introduction</LI><P></P><P><LI>Vector spaces </LI><P></P><P><LI>Normed linear spaces </LI><P></P><P><LI>Inner product spaces </LI><P></P><P><LI>Banach spaces </LI><P></P><P><LI>Hilbert spaces </LI><P></P><P><LI>Projection on finite dimensional spaces</LI><P></P><P><LI>Change of basis
  • Gram-Schmidt othogonalization process</LI><P></P><P><LI>Riesz bases and frame conditions</LI><P></P><P><LI>References for further reading</LI><P></P></OL></OL><OL><B><P></P></OL><P>4. Operators</P><OL><P></P></OL><OL><OL></B><P><LI>Mapping of sets, general concept of functions</LI><P></P><P><LI>Operators</LI><P></P><P><LI>Linear and adjoint operators</LI><P></P><P><LI>Functionals and dual space</LI><P></P><P><LI>Spectrum of bounded linear self-adjoint operator </LI><P></P><P><LI>Classification of differential operators</LI><P></P><P><LI>Existence, uniqueness and regularity of solution</LI><P></P><P><LI>References</LI><P></P></OL></OL><OL><B><P></P></OL><P>5. Theoretical foundations of the finite element method</B> </P><OL><P></P></OL><OL><OL><P><LI>Distribution theory</LI><P></P><P><LI>Sobolev spaces</LI><P></P><P><LI>Variational Method</LI><P></P><P><LI>Nonconforming elements and patch test</LI><P></P><P><LI>References for further reading</LI><P></P></OL></OL><OL><B><P></P></OL><P>6. Wavelet- based methods for differential equations</P><OL><P></P></OL><OL><OL></B><P><LI>Fundamentals of continuous and discrete wavelets</LI><P></P><P><LI>Multiscaling</LI><P></P><P><LI>Classification of wavelet basis functions </LI><P></P><P><LI>Discrete wavelet transform </LI><P></P><P><LI>Lifting scheme for discrete wavelet transform </LI><P></P><P><LI>Lifting scheme to customize wavelets </LI><P></P><P><LI>Non-standard form of matrix and its solution </LI><P></P><P><LI>Multigrid method</LI><P></P><P><LI>References for further reading</LI><P></P></OL></OL><OL><B><P></P></OL><P>7. Error
  • estimation</B></P><OL><P></P><OL><P><LI>Introduction</LI><P></P><I><P><LI>A-priori</I> error estimation</LI><P></P><P><LI>Recovery based error estimators </LI><P></P><P><LI>Residual based error estimators </LI><P></P><P><LI>Goal oriented error estimators</LI><P></P><P><LI>Hierarchical and wavelet based error estimator</LI><P></P><P><LI>References for further reading</LI><P></P></OL></OL><B><P>Appendix</B><I><I></P></I></I>