Row Scaling as a Preconditioner for Certain Nonsymmetric Linear Systems with Discontinuous Coefficients

Linear systems with large differences between coefficients, called “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 Lpnorm of its coefficients. This preconditioning is called geometric scaling (GS). GS is a particular form of a diagonal preconditioner. In the literature, diagonal scaling is usually applied to both sides of the system matrix in order to preserve symmetry and enable the use of the conjugate gradient (CG). This work is restricted to nonsymmetric systems.