Green's Functions For Preconditioning

Daniel Loghin


In this paper we look at the connection between Green's functions and preconditioners for systems of equations arising through the discretization of linear partial differential equations. We find that Green's functions, as generalized inverses of differential operators, can sometimes offer a useful insight into the properties of the discrete inverse operator. More precisely, their properties (singularity, decay etc.) translate into a particular matrix structure of the discrete inverse, which should be approximated by a good preconditioner. In the case of the advection-diffusion operator it is found that efficient preconditioning is achieved with block lower triangular matrices (including the Gauss-Seidel matrix) under a flow-oriented numbering of the variables.