Analysis of hierarchical SSOR for three dimensional isotropic model problem

TL;DR

This paper introduces a hierarchical SSOR method that enhances convergence speed for 3D isotropic problems and can serve as an efficient smoother in multigrid methods, outperforming traditional approaches.

Contribution

It presents a novel hierarchical SSOR method with proven faster convergence and lower costs, suitable as a standalone solver or smoother in multigrid algorithms.

Findings

HSSOR converges faster than ILU(0), SSOR, and Block SSOR.

HSSOR requires no additional storage or construction costs.

Eigenvalues and condition number expressions for HSSOR are derived.

Abstract

In this paper, we study a hierarchical SSOR (HSSOR) method which could be used as a standalone method or as a smoother for a two-grid method. It is found that the method leads to faster convergence compared to more costly incomplete LU (ILU(0)) with no fill-in, the SSOR, and the Block SSOR method. Moreover, for a two-grid method, numerical experiments suggests that HSSOR can be a better replacement for SSOR smoother both having no storage requirements and have no construction costs. Using Fourier analysis, ex- pressions for the eigenvalues and the condition number of HSSOR preconditioned problem is derived for the three-dimensional isotropic model problem.