Thesis: Use of Wavelet Transform-based tools for numerical solution of differential equations

[Victoria Vampa (victoriavampa@gmail.com)]

The aim of this thesis is the development of tools and strategies based on wavelet transform to solve differential equations.

At a first stage we analyze the feasibility and capability of using wavelet bases in the Finite Element method. In particular, for Mindlin-Reissner plate model, Daubechies Scaling Wavelet elements (DSWN) were designed. These elements can be easily constructed using independent interpolation of each displacement function. Due to orthonormality, compactly supported and nesting properties of the Daubechies wavelets, numerical results obtained with DSWN elements have very good accuracy and non-locking behavior.
Consequently, they are efficient for solving plate bending problems,
in both cases: thick and thin plates, and also when they are applied to problems having localized
singularities.

In a second part, which is the most original aspect of this thesis, a new method is proposed to solve a second order coercive boundary value problem using B-cubic-spline functions in an efficient manner.

This proposal combines variational equations with a collocation scheme using B-splines as scaling functions and yields an approximation at an initial scale. Bases of wavelets are designed in order to have a multiresolution structure on the interval. Then, a refinement process using wavelets is presented and convergence is proved. Bound of the approximation error are derived. Through the algorithm proposed, an improved solution
with minimal computational effort is obtained.

We present several numerical results showing the good behavior of the method. Approximate solutions are computed in scaling-spline form and improved with wavelets. Convergence of the solutions obtained using the proposed method are found to compare favorably to other numerical techniques.


keywords
Wavelet-finite element, beam element, plate element, B-spline functions, Multiresolution Analysis, Wavelet-Galerkin.

http://www.mate.unlp.edu.ar/tesis/tesis_VVampa.pdf