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. 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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.
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.