Preconditioners Based on Fundamental Solutions

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 E*(Pu) = u. Inspired by this, we choose the preconditioner to be a discretization of an approximate inverse K, given by a convolution-like operator with E as a kernel.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.

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