This thesis is concerned with the numerical solution of boundary integral equations
and the numerical analysis of iterative methods. In the first part, we assume
the boundary to be smooth in order to work with compact operators; while in the
second part we investigate the problem arising from allowing piecewise smooth
boundaries. Although in principle most results of the thesis apply to general problems
of reformulating boundary value problems as boundary integral equations
and their subsequent numerical solutions, we consider the Helmholtz equation
arising from acoustic problems as the main model problem.
In Chapter 1, we present the background material of reformulation of Helmhoitz
boundary value problems into boundary integral equations by either the indirect
potential method or the direct method using integral formulae. The problem of
ensuring unique solutions of integral equations for exterior problems is specifically
discussed. In Chapter 2, we discuss the useful numerical techniques for
solving second kind integral equations. In particular, we highlight the superconvergence
properties of iterated projection methods and the important procedure
of Nystrom interpolation.
In Chapter 3, the multigrid type methods as applied to smooth boundary
integral equations are studied. Using the residual correction principle, we are
able to propose some robust iterative variants modifying the existing methods to
seek efficient solutions. In Chapter 4, we concentrate on the conjugate gradient
method and establish its fast convergence as applied to the linear systems arising from general boundary element equations. For boundary integral equalisations on smooth boundaries we have observed, as the underlying mesh sizes decrease,
faster convergence of multigrid type methods and fixed step convergence of the
conjugate gradient method.
In the case of non-smooth integral boundaries, we first derive the singular
forms of the solution of boundary integral solutions for Dirichlet problems and
then discuss the numerical solution in Chapter 5. Iterative methods such as two
grid methods and the conjugate gradient method are successfully implemented
in Chapter 6 to solve the non-smooth integral equations. The study of two
grid methods in a general setting and also much of the results on the conjugate
gradient method are new. Chapters 3, 4 and 5 are partially based on publications
[4], [5] and [35] respectively.Department of Mathematics and Statistics,
Polytechnic South Wes
