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

Abstract: 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 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.

• The paper introduces geometric scaling as a preconditioner that scales equations by their Lp-norm to address coefficient disparities.
• The paper evaluates GS on nonsymmetric systems from convection-diffusion PDEs using iterative solvers like GMRES and Bi-CGSTAB, with and without ILUT.
• The paper shows that GS improves convergence rates by reducing the concentration of eigenvalues near the origin.

The paper proposes and evaluates a preconditioning technique known as geometric scaling (GS) for solving linear systems with discontinuous coefficients. Such systems frequently emerge in the context of partial differential equations (PDEs) that model physical phenomena in heterogeneous media. Traditional solutions leverage domain decomposition techniques, but these can be cumbersome to implement when faced with complicated geometries or unstructured grids.

Key Contributions

1. Introduction of Geometric Scaling (GS):
   • GS involves scaling the equations by dividing each equation by the Lp-norm of its coefficients.
   • This process belongs to a class of preconditioners and aims to address the disparity in coefficients, which can adversely affect convergence rates of iterative solvers.
2. Evaluation of GS:
   • The efficacy of the GS preconditioner was tested on a variety of nonsymmetric linear systems derived from convection-diffusion elliptic PDEs with small to moderate convection terms.
   • The study specifically focused on the performance of iterative solvers such as GMRES and Bi-CGSTAB, both with and without the ILUT preconditioner.
3. Improvement in Convergence:
   • Results indicated that the application of GS substantially improved the convergence rates of the iterative solvers.
   • GS was particularly effective in reducing the concentration of eigenvalues around the origin, thereby facilitating a more favorable distribution of eigenvalues for iterative methods.

Implications and Use Cases

• While diagonal scaling is a well-known technique for enhancing convergence properties in general contexts, the study emphasizes its specific utility in scenarios featuring discontinuous coefficients.
• The findings suggest that GS, due to its simplicity and effectiveness, could be a valuable tool in numerical simulations involving heterogeneous media, particularly when dealing with complex geometries or unstructured grids.

Conclusion

The introduction of geometric scaling as a preconditioner offers a promising approach for solving linear systems with discontinuous coefficients. By improving convergence rates and eigenvalue distributions, GS has the potential to significantly enhance the performance of standard iterative solvers in computational scenarios modeled by convection-diffusion elliptic PDEs.