This thesis is concerned with the solution of large systems
of linear algebraic equations in which the matrix of coefficients
is sparse. Such systems occur in the numerical solution of elliptic
partial differential equations by finite-difference methods. By
applying some well-known iterative methods, usually used to solve
linear PDE systems, the thesis investigates their applicability to
solve a set of four mildly nonlinear test problems.
In Chapter 4 we study the basic iterative methods and semiiterative
methods for linear systems. In particular, we derive
and apply the CS, SOR, SSOR methods and the SSOR method extrapolated
by the Chebyshev acceleration strategy.
In Chapter 5, three ways of accelerating the SOR method are
described together with the applications to the test problems.
Also the Newton-SOR method and the SOR-Newton method are derived
and applied to the same problems.
In Chapter 6, the Alternating Directions Implicit methods
are described. Two versions are studied in detail, namely, the
Peaceman-Rachford and the Douglas-Rachford methods. They have
been applied to the test problems for cycles of 1, 2 and 3
In Chapter 7, the conjugate gradients method and the conjugate
gradient acceleration procedure are described together with some
preconditioning techniques. Also an approximate LU-decomposition
algorithm (ALUBOT algorithm) is given and then applied in conjunction
with the Picard and Newton methods.
Chapter 8 contains the final conclusions.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.