Galerkin boundary element methods with multiwavelets

Rajaguru, PR

Abstract

In general the numerical solution of boundary integral equations arising from the reformulation
of partial differential equations leads to full coefficient matrices. The discrete
system can be solved in O(N2 ) operations by iterative solvers of the Conjugate
Gradient type possibly with a preconditioner. We are interested in fast methods such as
wavelets/mutiwavelets, that reduce the computational cost to O(N logp N) with p a small
positive integer. Wavelets are attractive for the numerical solution of integral equations because
their vanishing moments property leads to operator compression in the sense that the
resulting linear system has many elements which are negligible or very small. However, to
obtain wavelets with compact support and high order of vanishing moments, the length of
the support increases as the order of the vanishing moments 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 this thesis we are
concerned with multi wavelets. They have proved to be very efficient and effective basis
functions due to the fact that the coefficients of a multiwavelet expansion decay rapidly for
a large class of functions. In a multiwavelet method the unknown function in the boundary
integral equation is approximated by a finite number of mutiwavelet basis functions.
In Chapter 1 we review the methods and techniques required for reformulations of
boundary integral equations from PDE's, we also discuss how these boundary integral
equations may be discretised and discuss the solution process. In Chapter 2, we discuss
wavelet and multiwavelet bases and their characteristics. In Chapter 3, we consider the
boundary element method, namely, the standard Galerkin method with multiwavelet basis
functions. For this method two types of compression strategies are developed which only require the computation of the significant matrix elements. We show that there are
O(N\og N) such significant elements. In Chapter (4) we consider the boundary element
method, with the so called, the non-standard representation, using multiwavelet basis functions.
For this method also two types of compression strategies are developed which require
the computation of the significant matrix elements . In Chapter 5 we discuss briefly
the Galerkin boundary element methods with multiwavelets on nonsmooth boundary.

Citation

Rajaguru, P. Galerkin boundary element methods with multiwavelets. (Dissertation). University of Salford

• Thesis Type: Dissertation
• Deposit Date: Aug 19, 2021
• Additional Information: Funders : Engineering and Physical Sciences Research Council (EPSRC)
• Award Date: Jan 1, 2007