### Abstract

We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variablecoefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocitypressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565-7595], as well as established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

Original language | English (US) |
---|---|

Pages (from-to) | 1263-1297 |

Number of pages | 35 |

Journal | Communications in Computational Physics |

Volume | 16 |

Issue number | 5 |

DOIs | |

State | Published - Nov 1 2014 |

### Fingerprint

### Keywords

- GMRES
- Preconditioning
- Projection method
- Saddle point problems
- Stokes flow
- Variable density
- Variable viscosity

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Communications in Computational Physics*,

*16*(5), 1263-1297. https://doi.org/10.4208/cicp.070114.170614a

**Efficient variable-coefficient finite-volume stokes solvers.** / Cai, Mingchao; Nonaka, Andy; Bell, John B.; Griffith, Boyce E.; Donev, Aleksandar.

Research output: Contribution to journal › Article

*Communications in Computational Physics*, vol. 16, no. 5, pp. 1263-1297. https://doi.org/10.4208/cicp.070114.170614a

}

TY - JOUR

T1 - Efficient variable-coefficient finite-volume stokes solvers

AU - Cai, Mingchao

AU - Nonaka, Andy

AU - Bell, John B.

AU - Griffith, Boyce E.

AU - Donev, Aleksandar

PY - 2014/11/1

Y1 - 2014/11/1

N2 - We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variablecoefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocitypressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565-7595], as well as established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

AB - We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variablecoefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocitypressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565-7595], as well as established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

KW - GMRES

KW - Preconditioning

KW - Projection method

KW - Saddle point problems

KW - Stokes flow

KW - Variable density

KW - Variable viscosity

UR - http://www.scopus.com/inward/record.url?scp=84907209769&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907209769&partnerID=8YFLogxK

U2 - 10.4208/cicp.070114.170614a

DO - 10.4208/cicp.070114.170614a

M3 - Article

VL - 16

SP - 1263

EP - 1297

JO - Communications in Computational Physics

JF - Communications in Computational Physics

SN - 1815-2406

IS - 5

ER -