Analysis of Smoothing Matrices for the Preconditioning of Elliptic Difference Equations

Abstract
Smoothing techniques have been used for stabilizing explicit time integration of parabolic and hyperbolic initial‐boundary value problems. Similar techniques can be used for the preconditioning of elliptic difference equations. Such techniques are analysed in this paper. It is shown that the spectral radius of the Jacobian matrix associated with the system of equations can be reduced considerably by this type of preconditioners, without much computational effort. Theoretically, this results in a much more rapid convergence of function iteration methods like the Jacobi type methods. The use of smoothing techniques is illustrated for a few one‐dimensional and two‐dimensional problems, both of linear and nonlinear type. The numerical results show that the use of rather simple smoothing matrices reduces the number of iterations by at least a factor 10.

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