论狭小几何结构中斯托克斯方程的高效预处理

IF 1.8 3区 数学 Q1 MATHEMATICS
Vladislav Pimanov, Oleg Iliev, Ivan Oseledets, Ekaterina Muravleva
{"title":"论狭小几何结构中斯托克斯方程的高效预处理","authors":"Vladislav Pimanov, Oleg Iliev, Ivan Oseledets, Ekaterina Muravleva","doi":"10.1002/nla.2581","DOIUrl":null,"url":null,"abstract":"It is known (see, e.g., [SIAM J. Matrix Anal. Appl. 2014;35(1):143‐173]) that the performance of iterative methods for solving the Stokes problem essentially depends on the quality of the preconditioner for the Schur complement matrix, . In this paper, we consider two preconditioners for : the identity one and the SIMPLE one, and numerically study their performance for solving the Stokes problem in tight geometries. The latter are characterized by a high surface‐to‐volume ratio. We show that for such geometries, can become severely ill‐conditioned, having a very large condition number and a significant portion of non‐unit eigenvalues. As a consequence, the identity matrix, which is broadly used as a preconditioner for solving the Stokes problem in simple geometries, becomes very inefficient. We show that there is a correlation between the surface‐to‐volume ratio and the condition number of : the latter increases with the increase of the former. We show that the condition number of the diffusive SIMPLE‐preconditioned Schur complement matrix remains bounded when the surface‐to‐volume ratio increases, which explains the robust performance of this preconditioner for tight geometries. Further on, we use a direct method to calculate the full spectrum of and show that there is a correlation between the number of its non‐unit eigenvalues and the number of grid points at which no‐slip boundary conditions are prescribed. To illustrate the above findings, we examine the Pressure Schur Complement formulation for staggered finite‐difference discretization of the Stokes equations and solve it with the preconditioned conjugate gradient method. The practical problem which is of interest to us is computing the permeability of tight rocks.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the efficient preconditioning of the Stokes equations in tight geometries\",\"authors\":\"Vladislav Pimanov, Oleg Iliev, Ivan Oseledets, Ekaterina Muravleva\",\"doi\":\"10.1002/nla.2581\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It is known (see, e.g., [SIAM J. Matrix Anal. Appl. 2014;35(1):143‐173]) that the performance of iterative methods for solving the Stokes problem essentially depends on the quality of the preconditioner for the Schur complement matrix, . In this paper, we consider two preconditioners for : the identity one and the SIMPLE one, and numerically study their performance for solving the Stokes problem in tight geometries. The latter are characterized by a high surface‐to‐volume ratio. We show that for such geometries, can become severely ill‐conditioned, having a very large condition number and a significant portion of non‐unit eigenvalues. As a consequence, the identity matrix, which is broadly used as a preconditioner for solving the Stokes problem in simple geometries, becomes very inefficient. We show that there is a correlation between the surface‐to‐volume ratio and the condition number of : the latter increases with the increase of the former. We show that the condition number of the diffusive SIMPLE‐preconditioned Schur complement matrix remains bounded when the surface‐to‐volume ratio increases, which explains the robust performance of this preconditioner for tight geometries. Further on, we use a direct method to calculate the full spectrum of and show that there is a correlation between the number of its non‐unit eigenvalues and the number of grid points at which no‐slip boundary conditions are prescribed. To illustrate the above findings, we examine the Pressure Schur Complement formulation for staggered finite‐difference discretization of the Stokes equations and solve it with the preconditioned conjugate gradient method. The practical problem which is of interest to us is computing the permeability of tight rocks.\",\"PeriodicalId\":49731,\"journal\":{\"name\":\"Numerical Linear Algebra with Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-07-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Numerical Linear Algebra with Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1002/nla.2581\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Linear Algebra with Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/nla.2581","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

众所周知(参见,例如,[SIAM J. Matrix Anal. Appl.在本文中,我们考虑了两种前置条件器:同一前置条件器和 SIMPLE 前置条件器,并对它们在狭小几何形状中求解斯托克斯问题的性能进行了数值研究。后者的特点是高表面体积比。我们的研究表明,对于这种几何形状,斯托克斯矩阵的条件严重不足,具有非常大的条件数和大量非单位特征值。因此,被广泛用作解决简单几何形状中斯托克斯问题的前提条件的特征矩阵变得非常低效。我们发现,面体积比和条件数之间存在相关性:后者随着前者的增加而增加。我们证明,当表面与体积比增加时,扩散 SIMPLE 预处理舒尔补矩阵的条件数仍然是有界的,这解释了该预处理程序在狭小几何形状下的稳健性能。此外,我们使用直接方法计算了 Schur 补充矩阵的全谱,并证明其非单位特征值的数量与规定了无滑动边界条件的网格点数量之间存在相关性。为了说明上述发现,我们研究了交错有限差分离散斯托克斯方程的压力舒尔补全公式,并用预处理共轭梯度法求解。我们感兴趣的实际问题是计算致密岩石的渗透率。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the efficient preconditioning of the Stokes equations in tight geometries
It is known (see, e.g., [SIAM J. Matrix Anal. Appl. 2014;35(1):143‐173]) that the performance of iterative methods for solving the Stokes problem essentially depends on the quality of the preconditioner for the Schur complement matrix, . In this paper, we consider two preconditioners for : the identity one and the SIMPLE one, and numerically study their performance for solving the Stokes problem in tight geometries. The latter are characterized by a high surface‐to‐volume ratio. We show that for such geometries, can become severely ill‐conditioned, having a very large condition number and a significant portion of non‐unit eigenvalues. As a consequence, the identity matrix, which is broadly used as a preconditioner for solving the Stokes problem in simple geometries, becomes very inefficient. We show that there is a correlation between the surface‐to‐volume ratio and the condition number of : the latter increases with the increase of the former. We show that the condition number of the diffusive SIMPLE‐preconditioned Schur complement matrix remains bounded when the surface‐to‐volume ratio increases, which explains the robust performance of this preconditioner for tight geometries. Further on, we use a direct method to calculate the full spectrum of and show that there is a correlation between the number of its non‐unit eigenvalues and the number of grid points at which no‐slip boundary conditions are prescribed. To illustrate the above findings, we examine the Pressure Schur Complement formulation for staggered finite‐difference discretization of the Stokes equations and solve it with the preconditioned conjugate gradient method. The practical problem which is of interest to us is computing the permeability of tight rocks.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
3.40
自引率
2.30%
发文量
50
审稿时长
12 months
期刊介绍: Manuscripts submitted to Numerical Linear Algebra with Applications should include large-scale broad-interest applications in which challenging computational results are integral to the approach investigated and analysed. Manuscripts that, in the Editor’s view, do not satisfy these conditions will not be accepted for review. Numerical Linear Algebra with Applications receives submissions in areas that address developing, analysing and applying linear algebra algorithms for solving problems arising in multilinear (tensor) algebra, in statistics, such as Markov Chains, as well as in deterministic and stochastic modelling of large-scale networks, algorithm development, performance analysis or related computational aspects. Topics covered include: Standard and Generalized Conjugate Gradients, Multigrid and Other Iterative Methods; Preconditioning Methods; Direct Solution Methods; Numerical Methods for Eigenproblems; Newton-like Methods for Nonlinear Equations; Parallel and Vectorizable Algorithms in Numerical Linear Algebra; Application of Methods of Numerical Linear Algebra in Science, Engineering and Economics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信