### Abstract

The standard BDDC (balancing domain decomposition by constraints) preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model. This results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model; the cost is then often dominated by that of the problems on the subdomains. An important role is also played, both in theory and practice, by an averaging operator and in addition exact Dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains. The use of inexact solvers for these problems and even the replacement of the Dirichlet solvers by a trivial extension are considered. It is established that one of the resulting algorithms has the same eigenvalues as the standard BDDC algorithm, and the connection of another with the FETI-DP algorithm with a lumped preconditioner is also considered. Multigrid methods are used in the experimental work and under certain assumptions, it is established that the iteration count essentially remains the same as when exact solvers are used, while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear.

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

Pages (from-to) | 1415-1428 |

Number of pages | 14 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 196 |

Issue number | 8 |

DOIs | |

State | Published - Jan 20 2007 |

### Fingerprint

### Keywords

- BDDC
- Domain decomposition
- FETI-DP
- Inexact solvers
- Multi-level
- Multigrid
- Non-overlapping

### ASJC Scopus subject areas

- Computer Science Applications
- Computational Mechanics

### Cite this

*Computer Methods in Applied Mechanics and Engineering*,

*196*(8), 1415-1428. https://doi.org/10.1016/j.cma.2006.03.011

**On the use of inexact subdomain solvers for BDDC algorithms.** / Li, Jing; Widlund, Olof B.

Research output: Contribution to journal › Article

*Computer Methods in Applied Mechanics and Engineering*, vol. 196, no. 8, pp. 1415-1428. https://doi.org/10.1016/j.cma.2006.03.011

}

TY - JOUR

T1 - On the use of inexact subdomain solvers for BDDC algorithms

AU - Li, Jing

AU - Widlund, Olof B.

PY - 2007/1/20

Y1 - 2007/1/20

N2 - The standard BDDC (balancing domain decomposition by constraints) preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model. This results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model; the cost is then often dominated by that of the problems on the subdomains. An important role is also played, both in theory and practice, by an averaging operator and in addition exact Dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains. The use of inexact solvers for these problems and even the replacement of the Dirichlet solvers by a trivial extension are considered. It is established that one of the resulting algorithms has the same eigenvalues as the standard BDDC algorithm, and the connection of another with the FETI-DP algorithm with a lumped preconditioner is also considered. Multigrid methods are used in the experimental work and under certain assumptions, it is established that the iteration count essentially remains the same as when exact solvers are used, while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear.

AB - The standard BDDC (balancing domain decomposition by constraints) preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model. This results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model; the cost is then often dominated by that of the problems on the subdomains. An important role is also played, both in theory and practice, by an averaging operator and in addition exact Dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains. The use of inexact solvers for these problems and even the replacement of the Dirichlet solvers by a trivial extension are considered. It is established that one of the resulting algorithms has the same eigenvalues as the standard BDDC algorithm, and the connection of another with the FETI-DP algorithm with a lumped preconditioner is also considered. Multigrid methods are used in the experimental work and under certain assumptions, it is established that the iteration count essentially remains the same as when exact solvers are used, while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear.

KW - BDDC

KW - Domain decomposition

KW - FETI-DP

KW - Inexact solvers

KW - Multi-level

KW - Multigrid

KW - Non-overlapping

UR - http://www.scopus.com/inward/record.url?scp=33751102103&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33751102103&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2006.03.011

DO - 10.1016/j.cma.2006.03.011

M3 - Article

VL - 196

SP - 1415

EP - 1428

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

IS - 8

ER -