# Galerkin boundary element methods with multiwavelets

Rajaguru, PR 2007, Galerkin boundary element methods with multiwavelets , MPhil thesis, University of Salford.

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## 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.

Item Type: Thesis (MPhil) Amini, S (Supervisor) Schools > School of Computing, Science and Engineering Engineering and Physical Sciences Research Council (EPSRC) Institutional Repository 19 Aug 2021 08:41 27 Aug 2021 21:57 http://usir.salford.ac.uk/id/eprint/61639
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