### Abstract

Several domain decomposition methods of Neumann‐Neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi‐level methods. The Neumann‐Neumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. In its original form, however, the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases. In the new variants of the algorithms, considered in this paper, the preconditioners include global components, of low rank, to overcome this difficulty. Bounds are established for the condition number of the iteration operator, which are independent of the number of subregions, and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems. Results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions. ©1995 John Wiley & Sons, Inc.

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

Pages (from-to) | 121-155 |

Number of pages | 35 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 48 |

Issue number | 2 |

DOIs | |

State | Published - 1995 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*48*(2), 121-155. https://doi.org/10.1002/cpa.3160480203

**Schwarz methods of neumann‐neumann type for three‐dimensional elliptic finite element problems.** / Dryja, Maksymilian; Widlund, Olof B.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 48, no. 2, pp. 121-155. https://doi.org/10.1002/cpa.3160480203

}

TY - JOUR

T1 - Schwarz methods of neumann‐neumann type for three‐dimensional elliptic finite element problems

AU - Dryja, Maksymilian

AU - Widlund, Olof B.

PY - 1995

Y1 - 1995

N2 - Several domain decomposition methods of Neumann‐Neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi‐level methods. The Neumann‐Neumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. In its original form, however, the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases. In the new variants of the algorithms, considered in this paper, the preconditioners include global components, of low rank, to overcome this difficulty. Bounds are established for the condition number of the iteration operator, which are independent of the number of subregions, and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems. Results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions. ©1995 John Wiley & Sons, Inc.

AB - Several domain decomposition methods of Neumann‐Neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi‐level methods. The Neumann‐Neumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. In its original form, however, the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases. In the new variants of the algorithms, considered in this paper, the preconditioners include global components, of low rank, to overcome this difficulty. Bounds are established for the condition number of the iteration operator, which are independent of the number of subregions, and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems. Results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions. ©1995 John Wiley & Sons, Inc.

UR - http://www.scopus.com/inward/record.url?scp=84990712164&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990712164&partnerID=8YFLogxK

U2 - 10.1002/cpa.3160480203

DO - 10.1002/cpa.3160480203

M3 - Article

VL - 48

SP - 121

EP - 155

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 2

ER -