@TechReport{ it:2005-001,
author = {Henrik Brand{\'e}n and Per Sundqvist},
title = {Preconditioners Based on Fundamental Solutions},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2005},
number = {2005-001},
month = jan,
note = {Revised version of IT technical report 2000-032},
abstract = {We consider a new preconditioning technique for the
iterative solution of linear systems of equations that
arise when discretizing partial differential equations. The
method is applied to finite difference discretizations, but
the ideas apply to other discretizations too.
If $E$ is a fundamental solution of a differential operator
$P$, we have \mbox{$E\ast(Pu)=u$.} Inspired by this, we
choose the preconditioner to be a discretization of an
approximative inverse $K$, given by \[
(Ku)(x)=\int_{\Omega}E(x-y)u(y)dy, \qquad
x\in\Omega\subset\mathds{R}^d, \] where $\Omega$ is the
domain of interest.
We present analysis showing that if $P$ is a first order
differential operator, $KP$ is bounded, and numerical
results show grid independent convergence for first order
partial differential equations, using fixed point
iterations.
For the second order convection-diffusion equation
convergence is no longer grid independent when using fixed
point iterations, a result that is consistent with our
theory. However, if the grid is chosen to give a fixed
number of grid points within boundary layers, the number of
iterations is independent of the physical viscosity
parameter. Also, if GMRES is used together with the
proposed preconditioner, the numbers of iterations decrease
as the grid is refined, also for fixed viscosity.}
}