Galerkin boundary element methods with multiwavelets

In general the numerical solution of boundary integral equations arising from the reformulationof partial differential equations leads to full coefficient matrices. The discretesystem can be solved in O(N2 ) operations by iterative solvers of the ConjugateGradient type possibly with a preconditioner. We are interested in fast methods such aswavelets/mutiwavelets, that reduce the computational cost to O(N logp N) with p a smallpositive integer. Wavelets are attractive for the numerical solution of integral equations becausetheir vanishing moments property leads to operator compression in the sense that theresulting linear system has many elements which are negligible or very small. However, toobtain wavelets with compact support and high order of vanishing moments, the length ofthe support increases as the order of the vanishing moments increases. This causes difficultieswith the practical use of wavelets particularly at edges and corners. However, withmultiwavelets, an increase in the order of vanishing moments is obtained not by increasingthe support but by increasing the number of mother wavelets. In this thesis we areconcerned with multi wavelets. They have proved to be very efficient and effective basisfunctions due to the fact that the coefficients of a multiwavelet expansion decay rapidly fora large class of functions. In a multiwavelet method the unknown function in the boundaryintegral equation is approximated by a finite number of mutiwavelet basis functions.In Chapter 1 we review the methods and techniques required for reformulations ofboundary integral equations from PDE's, we also discuss how these boundary integralequations may be discretised and discuss the solution process. In Chapter 2, we discusswavelet and multiwavelet bases and their characteristics. In Chapter 3, we consider theboundary element method, namely, the standard Galerkin method with multiwavelet basisfunctions. For this method two types of compression strategies are developed which only require the computation of the significant matrix elements. We show that there areO(N\og N) such significant elements. In Chapter (4) we consider the boundary elementmethod, with the so called, the non-standard representation, using multiwavelet basis functions.For this method also two types of compression strategies are developed which requirethe computation of the significant matrix elements . In Chapter 5 we discuss brieflythe Galerkin boundary element methods with multiwavelets on nonsmooth boundary.

Level dc:type.qualificationlevel

Master's Level (Level 7)

Year dc:date.issued

2007

Author dc:creator

• Rajaguru, PR

Language dc:language

en

Identifier

oai:salford-repository.worktribe.com:1336980

OAI identifier oai:identifier

oai:salford-repository.worktribe.com:1336980