### Abstract

Overlap is essential for the classical Schwarz method to be convergent when solving elliptic problems. Over the last decade, it was however observed that when solving systems of hyperbolic partial differential equations, the classical Schwarz method can be convergent even without overlap. We show that the classical Schwarz method without overlap applied to the Cauchy-Riemann equations which represent the discretization in time of such a system, is equivalent to an optimized Schwarz method for a related elliptic problem, and thus must be convergent, since optimized Schwarz methods are well known to be convergent without overlap.

Original language | English |
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Title of host publication | Domain Decomposition Methods in Science and Engineering XVII |

Publisher | Springer |

Pages | 467-475 |

Number of pages | 9 |

Volume | 60 |

ISBN (Print) | 9783540751984 |

DOIs | |

Publication status | Published - 1 Dec 2008 |

Event | 17th International Conference on Domain Decomposition Methods - St. Wolfgang /Strobl, Austria Duration: 3 Jul 2006 → 7 Jul 2006 |

### Publication series

Name | Lecture Notes in Computational Science and Engineering |
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Volume | 60 |

ISSN (Print) | 1439-7358 |

### Conference

Conference | 17th International Conference on Domain Decomposition Methods |
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Country | Austria |

City | St. Wolfgang /Strobl |

Period | 3/07/06 → 7/07/06 |

### Keywords

- domain decomposition methods
- partial differential equations
- Cauchy-Riemann equations
- discretization
- elliptic problems
- hyperbolic partial differential equation
- hyperbolic system
- Schwarz method

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## Cite this

*Domain Decomposition Methods in Science and Engineering XVII*(Vol. 60, pp. 467-475). (Lecture Notes in Computational Science and Engineering; Vol. 60). Springer. https://doi.org/10.1007/978-3-540-75199-1_59