Optimized sparse approximate inverse smoothers for solving Laplacian linear systems

Abstract

In this paper we propose and analyze new efficient sparse approximate inverse (SAI) smoothers for solving the two-dimensional (2D) and three-dimensional (3D) Laplacian linear system with geometric multigrid methods. Local Fourier analysis shows that our proposed SAI smoother for 2D achieves a much smaller smoothing factor than the state-of-the-art SAI smoother studied in Bolten et al. (2016) [12]. The proposed SAI smoother for 3D cases provides smaller optimal smoothing factor than that of weighted Jacobi smoother. Numerical results validate our theoretical conclusions and illustrate the high-efficiency and high-effectiveness of our proposed SAI smoothers. Such SAI smoothers have the advantage of inherent parallelism.