Analysis of efficient preconditioner for solving Poisson equation with Dirichlet boundary condition in irregular three-dimensional domains

• Published in: Journal of computational physics 2024-12, Vol.519, p.113418, Article 113418
• Main Authors: Hwang, Geonho, Park, Yesom, Lee, Yueun, Kang, Myungjoo
• Format: Article
• Language: English
• Subjects: Analysis; Irregular domain; MILU preconditioner; Poisson equation; Three dimension
• ISSN: 0021-9991
• DOI: 10.1016/j.jcp.2024.113418

This paper analyzes modified ILU (MILU)-type preconditioners for efficiently solving the Poisson equation with Dirichlet boundary conditions on irregular domains. In [1], the second-order accuracy of a finite difference scheme developed by Gibou et al. [2] and the effect of the MILU preconditioner were presented for two-dimensional problems. However, the analyses do not directly extend to three-dimensional problems. In this paper, we first demonstrate that the Gibou method attains second-order convergence for three-dimensional irregular domains, yet the discretized Laplacian exhibits a condition number of O(h−2) for a grid size h. We show that the MILU preconditioner reduces the order of the condition number to O(h−1) in three dimensions. Furthermore, we propose a novel sectored-MILU preconditioner, defined by a sectorized lexicographic ordering along each axis of the domain. We demonstrate that this preconditioner reaches a condition number of order O(h−1) as well. Sectored-MILU not only achieves a similar or better condition number than conventional MILU but also improves parallel computing efficiency, enabling very efficient calculation when the dimensionality or problem size increases significantly. Our findings extend the feasibility of solving large-scale problems across a range of scientific and engineering disciplines. •This study extends the analysis of the Gibou method to irregular three-dimensional domains, demonstrating second-order convergence in irregular domains.•This section presents proof of the optimality of the MILU preconditioner, which reduces the condition number of the discretized system from O(h−2) to O(h−1).•It introduces and theoretically demonstrates the SMILU preconditioner with sectorized ordering, which improves parallel computing efficiency while maintaining optimal conditioning.•Comprehensive theoretical analyses and numerical experiments are conducted to confirm the theoretical analyses in both 2D and 3D scenarios.•This study contributes to the advancement of computational efficiency in solving large-scale Poisson problems in scientific and engineering domains.