Theory and applications of the multiwavelets for compression of boundary integral operators

Nixon, SP

Abstract

In general the numerical solution of boundary integral equations leads to full coefficient
matrices. The discrete system can be solved in O(N2) operations by iterative solvers of
the Conjugate Gradient type. Therefore, we are interested in fast methods such as fast
multipole and wavelets, that reduce the computational cost to O(N lnp N).
In this thesis we are concerned with wavelet methods. They have proved to be very
efficient and effective basis functions due to the fact that the coefficients of a wavelet expansion
decay rapidly for a large class of functions. Due to the multiresolution property
of wavelets they provide accurate local descriptions of functions efficiently. For example
in the presence of corners and edges, the functions can still be approximated with a linear
combination of just a few basis functions. Wavelets are attractive for the numerical
solution of integral equations because their vanishing moments property leads to operator
compression. However, to obtain wavelets with compact support and high order of vanishing
moments, the length of the support increases as the order of the vanishingmoments
increases. This causes difficulties with the practical use of wavelets particularly at edges
and corners. However, with multiwavelets, an increase in the order of vanishing moments
is obtained not by increasing the support but by increasing the number of mother wavelets.
In chapter 2 we review the methods and techniques required for these reformulations,
we also discuss how these boundary integral equations may be discretised by a boundary
element method. In chapter 3, we discuss wavelet and multiwavelet bases. In chapter
4, we consider two boundary element methods, namely, the standard and non-standard
Galerkin methods with multiwavelet basis functions. For both methods compression
strategies are developed which only require the computation of the significant matrix elements.
We show that they are O(N logp N) such significant elements. In chapters 5 and
6 we apply the standard and non-standard Galerkin methods to several test problems.

Citation

Nixon, S. Theory and applications of the multiwavelets for compression of boundary integral operators. (Thesis). University of Salford, UK

• Thesis Type: Thesis
• Deposit Date: Jun 8, 2009
• Publicly Available Date: Jun 8, 2009
• Additional Information: Additional Information : PhD supervisor: Professor Sia Amini
• Award Date: Jun 1, 2004