### Abstract

Multi-trace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. The first aim of the present contribution is to provide a gentle introduction to MTFs. We introduce these formulations on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multi-trace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calderón projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments.

Language | English |
---|---|

Pages | 69-86 |

Number of pages | 18 |

Journal | Applied Numerical Mathematics |

Volume | 135 |

Early online date | 24 Jul 2018 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Fingerprint

### Keywords

- multi-trace formulations
- Calderon projectors
- Dirichlet to Neumann operators
- optimal Schwarz methods

### Cite this

*Applied Numerical Mathematics*,

*135*, 69-86. https://doi.org/10.1016/j.apnum.2018.07.006

}

*Applied Numerical Mathematics*, vol. 135, pp. 69-86. https://doi.org/10.1016/j.apnum.2018.07.006

**An introduction to multitrace formulations and associated domain decomposition solvers.** / Claeys, X.; Dolean, V.; Gander, M. J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An introduction to multitrace formulations and associated domain decomposition solvers

AU - Claeys, X.

AU - Dolean, V.

AU - Gander, M. J.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Multi-trace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. The first aim of the present contribution is to provide a gentle introduction to MTFs. We introduce these formulations on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multi-trace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calderón projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments.

AB - Multi-trace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. The first aim of the present contribution is to provide a gentle introduction to MTFs. We introduce these formulations on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multi-trace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calderón projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments.

KW - multi-trace formulations

KW - Calderon projectors

KW - Dirichlet to Neumann operators

KW - optimal Schwarz methods

U2 - 10.1016/j.apnum.2018.07.006

DO - 10.1016/j.apnum.2018.07.006

M3 - Article

VL - 135

SP - 69

EP - 86

JO - Applied Numerical Mathematics

T2 - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -