Title: | 2-dimensional primal domain decomposition theory in detail (English) |

Author: | Lukáš, Dalibor |

Author: | Bouchala, Jiří |

Author: | Vodstrčil, Petr |

Author: | Malý, Lukáš |

Language: | English |

Journal: | Applications of Mathematics |

ISSN: | 0862-7940 (print) |

ISSN: | 1572-9109 (online) |

Volume: | 60 |

Issue: | 3 |

Year: | 2015 |

Pages: | 265-283 |

Summary lang: | English |

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Category: | math |

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Summary: | We give details of the theory of primal domain decomposition (DD) methods for a 2-dimensional second order elliptic equation with homogeneous Dirichlet boundary conditions and jumping coefficients. The problem is discretized by the finite element method. The computational domain is decomposed into triangular subdomains that align with the coefficients jumps. We prove that the condition number of the vertex-based DD preconditioner is $O((1+\log (H/h))^2)$, independently of the coefficient jumps, where $H$ and $h$ denote the discretization parameters of the coarse and fine triangulations, respectively. Although this preconditioner and its analysis date back to the pioneering work J. H. Bramble, J. E. Pasciak, A. H. Schatz (1986), and it was revisited and extended by many authors including M. Dryja, O. B. Widlund (1990) and A. Toselli, O. B. Widlund (2005), the theory is hard to understand and some details, to our best knowledge, have never been published. In this paper we present all the proofs in detail by means of fundamental calculus. (English) |

Keyword: | domain decomposition method |

Keyword: | finite element method |

Keyword: | preconditioning |

MSC: | 65F08 |

MSC: | 65N30 |

MSC: | 65N55 |

idZBL: | Zbl 06486911 |

idMR: | MR3419962 |

DOI: | 10.1007/s10492-015-0095-5 |

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Date available: | 2015-05-15T07:37:24Z |

Last updated: | 2020-07-02 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/144263 |

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Reference: | [1] Bramble, J. H., Pasciak, J. E., Schatz, A. H.: The construction of preconditioners for elliptic problems by substructuring. I.Math. Comput. 47 (1986), 103-134. Zbl 0615.65112, MR 0842125, 10.1090/S0025-5718-1986-0842125-3 |

Reference: | [2] Dryja, M., Smith, B. F., Widlund, O. B.: Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions.SIAM J. Numer. Anal. 31 (1994), 1662-1694. Zbl 0818.65114, MR 1302680, 10.1137/0731086 |

Reference: | [3] Dryja, M., Widlund, O. B.: Some domain decomposition algorithms for elliptic problems.Iterative Methods for Large Linear Systems Austin, TX, 1988. Academic Press, Boston 273-291 (1990). MR 1038100 |

Reference: | [4] Farhat, C., Roux, F.-X.: A method of finite element tearing and interconnecting and its parallel solution algorithm.Int. J. Numer. Methods Eng. 32 (1991), 1205-1227. Zbl 0758.65075, 10.1002/nme.1620320604 |

Reference: | [5] George, A.: Nested dissection of a regular finite element mesh.SIAM J. Numer. Anal. 10 (1973), 345-363. Zbl 0259.65087, MR 0388756, 10.1137/0710032 |

Reference: | [6] Mandel, J., Brezina, M.: Balancing domain decomposition for problems with large jumps in coefficients.Math. Comput. 65 (1996), 1387-1401. Zbl 0853.65129, MR 1351204, 10.1090/S0025-5718-96-00757-0 |

Reference: | [7] Mandel, J., Tezaur, R.: Convergence of a substructuring method with Lagrange multipliers.Numer. Math. 73 (1996), 473-487. Zbl 0880.65087, MR 1393176, 10.1007/s002110050201 |

Reference: | [8] Payne, L. E., Weinberger, H. F.: An optimal Poincaré inequality for convex domains.Arch. Ration. Mech. Anal. 5 (1960), 286-292. Zbl 0099.08402, MR 0117419, 10.1007/BF00252910 |

Reference: | [9] Toselli, A., Widlund, O.: Domain Decomposition Methods---Algorithms and Theory.Springer Series in Computational Mathematics 34 Springer, Berlin (2005). Zbl 1069.65138, MR 2104179 |

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