### Abstract

Balancing domain decomposition by constraints (BDDC) algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with Gauss-Lobatto-Legendre spectral elements in three dimensions. Initially mixed spectral elements are employed to discretize the almost incompressible elasticity system, but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped. Appropriate sets of primal constraints can be associated with the subdomain vertices, edges, and faces so that the resulting BDDC methods have a fast convergence rate independent of the almost incompressibility of the material. In particular, the condition number of the BDDC preconditioned operator is shown to depend only weakly on the polynomial degree n, the ratio H/h of subdomain and element diameters, and the inverse of the inf-sup constants of the subdomains and the underlying mixed formulation, while being scalable, i.e., independent of the number of subdomains and robust, i.e., independent of the Poisson ratio and Young's modulus of the material considered. These results also apply to the related dual-primal finite element tearing and interconnect (FETI-DP) algorithms defined by the same set of primal constraints. Numerical experiments, carried out on parallel computing systems, confirm these results.

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

Pages (from-to) | 3604-3626 |

Number of pages | 23 |

Journal | SIAM Journal on Scientific Computing |

Volume | 32 |

Issue number | 6 |

DOIs | |

State | Published - 2010 |

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### Keywords

- Almost incompressible elasticity
- Balancing domain decomposition by constraints preconditioners
- Domain decomposition
- Mixed spectral elements

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*32*(6), 3604-3626. https://doi.org/10.1137/100791701

**BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions.** / Pavarino, Luca F.; Widlund, Olof B.; Zampini, Stefano.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 32, no. 6, pp. 3604-3626. https://doi.org/10.1137/100791701

}

TY - JOUR

T1 - BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions

AU - Pavarino, Luca F.

AU - Widlund, Olof B.

AU - Zampini, Stefano

PY - 2010

Y1 - 2010

N2 - Balancing domain decomposition by constraints (BDDC) algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with Gauss-Lobatto-Legendre spectral elements in three dimensions. Initially mixed spectral elements are employed to discretize the almost incompressible elasticity system, but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped. Appropriate sets of primal constraints can be associated with the subdomain vertices, edges, and faces so that the resulting BDDC methods have a fast convergence rate independent of the almost incompressibility of the material. In particular, the condition number of the BDDC preconditioned operator is shown to depend only weakly on the polynomial degree n, the ratio H/h of subdomain and element diameters, and the inverse of the inf-sup constants of the subdomains and the underlying mixed formulation, while being scalable, i.e., independent of the number of subdomains and robust, i.e., independent of the Poisson ratio and Young's modulus of the material considered. These results also apply to the related dual-primal finite element tearing and interconnect (FETI-DP) algorithms defined by the same set of primal constraints. Numerical experiments, carried out on parallel computing systems, confirm these results.

AB - Balancing domain decomposition by constraints (BDDC) algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with Gauss-Lobatto-Legendre spectral elements in three dimensions. Initially mixed spectral elements are employed to discretize the almost incompressible elasticity system, but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped. Appropriate sets of primal constraints can be associated with the subdomain vertices, edges, and faces so that the resulting BDDC methods have a fast convergence rate independent of the almost incompressibility of the material. In particular, the condition number of the BDDC preconditioned operator is shown to depend only weakly on the polynomial degree n, the ratio H/h of subdomain and element diameters, and the inverse of the inf-sup constants of the subdomains and the underlying mixed formulation, while being scalable, i.e., independent of the number of subdomains and robust, i.e., independent of the Poisson ratio and Young's modulus of the material considered. These results also apply to the related dual-primal finite element tearing and interconnect (FETI-DP) algorithms defined by the same set of primal constraints. Numerical experiments, carried out on parallel computing systems, confirm these results.

KW - Almost incompressible elasticity

KW - Balancing domain decomposition by constraints preconditioners

KW - Domain decomposition

KW - Mixed spectral elements

UR - http://www.scopus.com/inward/record.url?scp=79251513986&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79251513986&partnerID=8YFLogxK

U2 - 10.1137/100791701

DO - 10.1137/100791701

M3 - Article

VL - 32

SP - 3604

EP - 3626

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 6

ER -