### Abstract

Balancing Neumann-Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant-pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing Neumann-Neumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and a factor that depends on the inverse of the inf-sup constants of the discrete problem and of the coarse sub-problem. Numerical results show that the method is quite fast; they are also fully consistent with the theory.

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

Pages (from-to) | 302-335 |

Number of pages | 34 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 55 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2002 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*55*(3), 302-335. https://doi.org/10.1002/cpa.10020

**Balancing Neumann-Neumann Methods for Incompressible Stokes Equations.** / Pavarino, Luca F.; Widlund, Olof B.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 55, no. 3, pp. 302-335. https://doi.org/10.1002/cpa.10020

}

TY - JOUR

T1 - Balancing Neumann-Neumann Methods for Incompressible Stokes Equations

AU - Pavarino, Luca F.

AU - Widlund, Olof B.

PY - 2002/3

Y1 - 2002/3

N2 - Balancing Neumann-Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant-pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing Neumann-Neumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and a factor that depends on the inverse of the inf-sup constants of the discrete problem and of the coarse sub-problem. Numerical results show that the method is quite fast; they are also fully consistent with the theory.

AB - Balancing Neumann-Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant-pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing Neumann-Neumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and a factor that depends on the inverse of the inf-sup constants of the discrete problem and of the coarse sub-problem. Numerical results show that the method is quite fast; they are also fully consistent with the theory.

UR - http://www.scopus.com/inward/record.url?scp=0036112010&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036112010&partnerID=8YFLogxK

U2 - 10.1002/cpa.10020

DO - 10.1002/cpa.10020

M3 - Article

AN - SCOPUS:0036112010

VL - 55

SP - 302

EP - 335

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 3

ER -