### Abstract

In the theory for domain decomposition algorithms of the iterative substructuring family, each subdomain is typically assumed to be the union of a few coarse triangles or tetrahedra. This is an unrealistic assumption, in particular if the subdomains result from the use of a mesh partitioner, in which case they might not even have uniformly Lipschitz continuous boundaries. The purpose of this study is to derive bounds for the condition number of these preconditioned conjugate gradient methods which depend only on a parameter in an isoperimetric inequality, two geometric parameters characterizing John and uniform domains, and the maximum number of edges of any subdomain. A related purpose is to explore to what extent well-known technical tools previously developed for quite regular subdomains can be extended to much more irregular subdomains. Some of these results are valid for any John domain, while an extension theorem, which is needed in this study, requires that the subdomains have complements which are uniform. The results, so far, are complete only for problems in two dimensions. Details are worked out for a FETI-DP algorithm and numerical results support the findings. Some of the numerical experiments illustrate that care must be taken when selecting the scaling of the preconditioners in the case of irregular subdomains.

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

Pages (from-to) | 2484-2504 |

Number of pages | 21 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 46 |

Issue number | 5 |

DOIs | |

State | Published - 2008 |

### Fingerprint

### Keywords

- Domain decomposition
- Dual-primal FETI
- Fractal subdomains
- Iterative substructures
- John and uniform domains
- Preconditioned

### ASJC Scopus subject areas

- Numerical Analysis

### Cite this

*SIAM Journal on Numerical Analysis*,

*46*(5), 2484-2504. https://doi.org/10.1137/070688675

**An analysis of a FETI-DP algorithm on irregular subdomains in the plane.** / Klawonn, Axel; Rheinbach, Oliver; Widlund, Olof B.

Research output: Contribution to journal › Article

*SIAM Journal on Numerical Analysis*, vol. 46, no. 5, pp. 2484-2504. https://doi.org/10.1137/070688675

}

TY - JOUR

T1 - An analysis of a FETI-DP algorithm on irregular subdomains in the plane

AU - Klawonn, Axel

AU - Rheinbach, Oliver

AU - Widlund, Olof B.

PY - 2008

Y1 - 2008

N2 - In the theory for domain decomposition algorithms of the iterative substructuring family, each subdomain is typically assumed to be the union of a few coarse triangles or tetrahedra. This is an unrealistic assumption, in particular if the subdomains result from the use of a mesh partitioner, in which case they might not even have uniformly Lipschitz continuous boundaries. The purpose of this study is to derive bounds for the condition number of these preconditioned conjugate gradient methods which depend only on a parameter in an isoperimetric inequality, two geometric parameters characterizing John and uniform domains, and the maximum number of edges of any subdomain. A related purpose is to explore to what extent well-known technical tools previously developed for quite regular subdomains can be extended to much more irregular subdomains. Some of these results are valid for any John domain, while an extension theorem, which is needed in this study, requires that the subdomains have complements which are uniform. The results, so far, are complete only for problems in two dimensions. Details are worked out for a FETI-DP algorithm and numerical results support the findings. Some of the numerical experiments illustrate that care must be taken when selecting the scaling of the preconditioners in the case of irregular subdomains.

AB - In the theory for domain decomposition algorithms of the iterative substructuring family, each subdomain is typically assumed to be the union of a few coarse triangles or tetrahedra. This is an unrealistic assumption, in particular if the subdomains result from the use of a mesh partitioner, in which case they might not even have uniformly Lipschitz continuous boundaries. The purpose of this study is to derive bounds for the condition number of these preconditioned conjugate gradient methods which depend only on a parameter in an isoperimetric inequality, two geometric parameters characterizing John and uniform domains, and the maximum number of edges of any subdomain. A related purpose is to explore to what extent well-known technical tools previously developed for quite regular subdomains can be extended to much more irregular subdomains. Some of these results are valid for any John domain, while an extension theorem, which is needed in this study, requires that the subdomains have complements which are uniform. The results, so far, are complete only for problems in two dimensions. Details are worked out for a FETI-DP algorithm and numerical results support the findings. Some of the numerical experiments illustrate that care must be taken when selecting the scaling of the preconditioners in the case of irregular subdomains.

KW - Domain decomposition

KW - Dual-primal FETI

KW - Fractal subdomains

KW - Iterative substructures

KW - John and uniform domains

KW - Preconditioned

UR - http://www.scopus.com/inward/record.url?scp=55349129284&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=55349129284&partnerID=8YFLogxK

U2 - 10.1137/070688675

DO - 10.1137/070688675

M3 - Article

VL - 46

SP - 2484

EP - 2504

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 5

ER -