Preconditioning for sparse matrices with applications

Mathematical models play an ever increasing role in the development of science and technology, as they offer the possibility to simulate, understand and optimize reality in computing laboratory. They can be used in a large variety of applications, e.g. fluid flows and air flows, mass and temperature distributions, pollution of air and water, electrical and gravitational interactions, to name only a few. Since physical experiments are often very difficult and costly, physical problems are often represented by partial differential equations which have to be solved by discrete numerical methods. The computation of an accurate approximation to the original continuous problem will usually lead to systems of linear equations in which the number of unknowns can be very large: currently, for many practical three-dimensional applications, the size of the linear systems is typically of the order of 1-100 million unknowns. Fortunately, the coefficient matrices arising in such computations are very sparse, i.e, only a very small part of the entries of A differs from zero. In this thesis, we consider the solution of x from the large sparse system of linear equations Ax = b for a real non-singular N × N matrix A and a given vector b.