### Abstract

In this paper, we provide techniques for the development and analysis of parallel multilevel preconditioners for the discrete systems which arise in numerical approximation of symmetric elliptic boundary value problems. These preconditioners are defined as a sum of independent operators on a sequence of nested subspaces of the full approximation space. On a parallel computer, the evaluation of these operators and hence of the preconditioner on a given function can be computed concurrently. We shall study this new technique for developing preconditioners first in an abstract setting, next by considering applications to second-order elliptic problems, and finally by providing numerically computed condition numbers for the resulting preconditioned systems. The abstract theory gives estimates on the condition number in terms of three assumptions. These assumptions can be verified for quasi-uniform as well as refined meshes in any number of dimensions. Numerical results for the condition number of the preconditioned systems are provided for the new algorithms and compared with other wellknown multilevel approaches.

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

Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Mathematics of Computation |

Volume | 55 |

Issue number | 191 |

DOIs | |

State | Published - Jul 1990 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*55*(191), 1-22. https://doi.org/10.1090/S0025-5718-1990-1023042-6

}

*Mathematics of Computation*, vol. 55, no. 191, pp. 1-22. https://doi.org/10.1090/S0025-5718-1990-1023042-6

**Parallel multilevel preconditioners.** / Bramble, James H.; Pasciak, Joseph E.; Xu, Jinchao.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Parallel multilevel preconditioners

AU - Bramble, James H.

AU - Pasciak, Joseph E.

AU - Xu, Jinchao

PY - 1990/7

Y1 - 1990/7

N2 - In this paper, we provide techniques for the development and analysis of parallel multilevel preconditioners for the discrete systems which arise in numerical approximation of symmetric elliptic boundary value problems. These preconditioners are defined as a sum of independent operators on a sequence of nested subspaces of the full approximation space. On a parallel computer, the evaluation of these operators and hence of the preconditioner on a given function can be computed concurrently. We shall study this new technique for developing preconditioners first in an abstract setting, next by considering applications to second-order elliptic problems, and finally by providing numerically computed condition numbers for the resulting preconditioned systems. The abstract theory gives estimates on the condition number in terms of three assumptions. These assumptions can be verified for quasi-uniform as well as refined meshes in any number of dimensions. Numerical results for the condition number of the preconditioned systems are provided for the new algorithms and compared with other wellknown multilevel approaches.

AB - In this paper, we provide techniques for the development and analysis of parallel multilevel preconditioners for the discrete systems which arise in numerical approximation of symmetric elliptic boundary value problems. These preconditioners are defined as a sum of independent operators on a sequence of nested subspaces of the full approximation space. On a parallel computer, the evaluation of these operators and hence of the preconditioner on a given function can be computed concurrently. We shall study this new technique for developing preconditioners first in an abstract setting, next by considering applications to second-order elliptic problems, and finally by providing numerically computed condition numbers for the resulting preconditioned systems. The abstract theory gives estimates on the condition number in terms of three assumptions. These assumptions can be verified for quasi-uniform as well as refined meshes in any number of dimensions. Numerical results for the condition number of the preconditioned systems are provided for the new algorithms and compared with other wellknown multilevel approaches.

UR - http://www.scopus.com/inward/record.url?scp=84966234195&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966234195&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1990-1023042-6

DO - 10.1090/S0025-5718-1990-1023042-6

M3 - Article

AN - SCOPUS:84966234195

VL - 55

SP - 1

EP - 22

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 191

ER -