@TechReport{	  it:2000-032,
  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		= {2000},
  number	= {2000-032},
  month		= nov,
  note		= {Revised version available as IT technical report
		  2005-001},
  abstract	= {We consider a new convergence acceleration 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 and the basic theory 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 the
		  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. The operator $K$ is only an
		  approximation of $P^{-1}$ since we do not integrate over
		  all of $\mathds{R}^d$ as for the convolution, and since we
		  impose boundary conditions. Two main advantages of this
		  method are that we can perform analysis before we
		  discretize the operators, and that there is a fast way of
		  applying the preconditioner using FFT.
		  
		  We present analysis showing that if $P$ is a first order
		  differential operator, $KP$ is bounded. The analysis also
		  describes how $K$ differs from $P^{-1}$. Implementation
		  aspects are considered, and numerical results show grid
		  independent convergence for first order partial
		  differential equations. For the second order
		  convection-diffusion equation convergence is no longer grid
		  independent, 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. }
}