Volume 14, Issue 4
On Direct and Semi-Direct Inverse of Stokes, Helmholtz and Laplacian Operators in View of Time-Stepper-Based Newton and Arnoldi Solvers in Incompressible CFD

H. Vitoshkin & A. Yu. Gelfgat

Commun. Comput. Phys., 14 (2013), pp. 1103-1119.

Published online: 2013-10

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  • Abstract

Factorization of the incompressible Stokes operator linking pressure and velocity is revisited. The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown, additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed.

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@Article{CiCP-14-1103, author = {}, title = {On Direct and Semi-Direct Inverse of Stokes, Helmholtz and Laplacian Operators in View of Time-Stepper-Based Newton and Arnoldi Solvers in Incompressible CFD}, journal = {Communications in Computational Physics}, year = {2013}, volume = {14}, number = {4}, pages = {1103--1119}, abstract = {

Factorization of the incompressible Stokes operator linking pressure and velocity is revisited. The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown, additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.300412.010213a}, url = {http://global-sci.org/intro/article_detail/cicp/7195.html} }
TY - JOUR T1 - On Direct and Semi-Direct Inverse of Stokes, Helmholtz and Laplacian Operators in View of Time-Stepper-Based Newton and Arnoldi Solvers in Incompressible CFD JO - Communications in Computational Physics VL - 4 SP - 1103 EP - 1119 PY - 2013 DA - 2013/10 SN - 14 DO - http://doi.org/10.4208/cicp.300412.010213a UR - https://global-sci.org/intro/article_detail/cicp/7195.html KW - AB -

Factorization of the incompressible Stokes operator linking pressure and velocity is revisited. The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown, additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed.

H. Vitoshkin & A. Yu. Gelfgat. (2020). On Direct and Semi-Direct Inverse of Stokes, Helmholtz and Laplacian Operators in View of Time-Stepper-Based Newton and Arnoldi Solvers in Incompressible CFD. Communications in Computational Physics. 14 (4). 1103-1119. doi:10.4208/cicp.300412.010213a
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