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