@PhDThesis{ itlic:2003-001,
author = {Per Sundqvist},
title = {Preconditioners and Fundamental Solutions},
school = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2003},
number = {2003-001},
type = {Licentiate thesis},
month = mar,
abstract = {New preconditioning techniques for the iterative solution
of systems of equations arising from discretizations of
partial differential equations are considered. Fundamental
solutions, both of differential and difference operators,
are used as kernels in discrete, truncated convolution
operators. The intention is to approximate inverses of
difference operators that arise when discretizing the
differential equations. The approximate inverses are used
as preconditioners.
The technique using fundamental solutions of differential
operators is applied to scalar problems in two dimensions,
and grid independent convergence is obtained for a first
order differential equation.
The problem of computing fundamental solutions of
difference operators is considered, and we propose a new
algorithm. It can be used also when the symbol of the
difference operator is not invertible everywhere, and it is
applicable in two or more dimensions.
Fundamental solutions of difference operators are used to
construct preconditioners for non-linear systems of
difference equations in two dimensions. Grid independent
convergence is observed for two standard finite difference
discretizations of the Euler equations in a
non-axisymmetric duct.}
}