Uppsala University
Department of Information Technology

### Technical Report 2005-001

# Preconditioners Based on Fundamental Solutions

### Henrik Brandén and Per Sundqvist

January 2005

**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 *E*(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, x∈Omegasubsetmathds{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.

*Note:* Revised version of IT technical report 2000-032

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