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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:523961 |
| Date | January 2004 |
| Creators | Nixon, Steven Paul |
| Publisher | University of Salford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://usir.salford.ac.uk/2035/ |
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