Geometric scaling: a simple preconditioner for certain linear systems with discontinuous coefficients

Linear systems with large differences between coefficients (“discontinuous coefficients”) arise in many cases in which partial differential equations (PDEs) model physical phenomena involving heterogeneous media. The standard approach to solving such problems is to use domain decomposition (DD) techniques, with domain boundaries conforming to the boundaries between the different media. This approach can be difficult to implement when the geometry of the domain boundaries is complicated or the grid is unstructured. This work examines the simple preconditioning technique of scaling the equations by dividing each equation by the Lp-norm of its coefficients. This preconditioning is called geometric scaling (GS). It has long been known that diagonal scaling can be useful in improving convergence, but there is no study on the general usefulness of this approach for discontinuous coefficients. GS was tested on several nonsymmetric linear systems with discontinuous coefficients derived from convection-diffusion elliptic PDEs with small to moderate convection terms. It is shown that GS improved the convergence properties of restarted GMRES and Bi-CGSTAB, with and without the ILUT preconditioner. GS was also shown to improve the distribution of the eigenvalues by reducing their concentration around the origin very significantly.