Optimal preconditioners on solving the Poisson equation with Neumann boundary conditions
- 주제(키워드) Poisson equation , Neumann boundary condition , Preconditioner , Modified ILU , Optimality , Fluid simulation
- 주제(기타) Computer Science, Interdisciplinary Applications
- 주제(기타) Physics, Mathematical
- 관리정보기술 faculty
- 등재 SCIE, SCOPUS
- 발행기관 ACADEMIC PRESS INC ELSEVIER SCIENCE
- 발행년도 2021
- 총서유형 Journal
- URI http://www.dcollection.net/handler/ewha/000000182530
- 본문언어 영어
- Published As http://dx.doi.org/10.1016/j.jcp.2021.110189
초록/요약
MILU preconditioner is well known [16,3] to be the optimal choice among all the ILU-type preconditioners in solving the Poisson equation with Dirichlet boundary conditions. However, it is less known which is an optimal preconditioner in solving the Poisson equation with Neumann boundary conditions. The condition number of an unpreconditioned matrix is as large as O (h(-2)), where h is the step size of grid. Only the optimal preconditioner results in condition number O (h(-1)), while the others such as Jacobi and ILU result in O (h(-2)). We review Relaxed ILU and Perturbed MILU preconditioners in the case of Neumann boundary conditions, and present empirical results which indicate that the former is optimal in two dimensions and the latter is optimal in two and three dimensions. To the best of our knowledge, these empirical results have not been rigorously verified yet. We present a formal proof for the optimality of Relaxed ILU in rectangular domains, and discuss its possible extension to general smooth domains and Perturbed MILU. (C) 2021 Elsevier Inc. All rights reserved.
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