Preconditioners and fundamental solutions

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