Mathematical Theory of Subdivision

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

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Hierarchical Basis Function;Wavelet Galerkin Method;functional analysis;Finite Difference Method;numerical analysis;Local Lipschitz Property;computer programming;Scaling Functions;partial differential equations;Haar Scaling Functions;Hilbert space;Haar Wavelets;wavelets-Galerkin methods;Finite Element Basis Function;Sobolev spaces;Bar Element;Filter Coefficients;Normed Linear Space;Daubechies Scaling Function;Linearly Independent;Lifting Scheme;Linear Vector Space;Element Stiffness Matrix;Discrete Wavelet Transform;Wavelet Solution;End End End;Fem Solution;Finite Element Solution;Exact Strain;Connection Coefficients;Cauchy Sequence