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A strongly convergent primal–dual method for nonoverlapping domain decomposition

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Abstract

We propose a primal–dual parallel proximal splitting method for solving domain decomposition problems for partial differential equations. The problem is formulated via minimization of energy functions on the subdomains with coupling constraints which model various properties of the solution at the interfaces. The proposed method can handle a wide range of linear and nonlinear problems, with flexible, possibly nonlinear, transmission conditions across the interfaces. Strong convergence in the energy spaces is established in this general setting, and without any additional assumption on the energy functions or the geometry of the problem. Several examples are presented.

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Authors and Affiliations

  1. UMR 5149, Institut de Mathématiques et de Modélisation de Montpellier, Université Montpellier II, 34095, Montpellier, France

    Hédy Attouch

  2. Departamento de Matemática, Universidad Técnica Federico Santa María, Santiago, Chile

    Luis M. Briceño-Arias

  3. Laboratoire Jacques-Louis Lions, UMR 7598, Sorbonne Universités, UPMC Univ. Paris 06, 75005, Paris, France

    Patrick L. Combettes

Authors
  1. Hédy Attouch
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  2. Luis M. Briceño-Arias
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  3. Patrick L. Combettes
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Corresponding author

Correspondence to Patrick L. Combettes.

Additional information

The work of H. Attouch was supported by ECOS under Grant C13E03, and by Air Force Office of Scientific Research, USAF, under Grant FA9550-14-1-0056. The work of L. M. Briceño-Arias and P. L. Combettes was supported by MathAmSud under Grant N13MATH01. L. M. Briceño-Arias was also supported by Conicyt under Grants Fondecyt 3120054 and 11140360, and under Grant Anillo ACT1106.

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Attouch, H., Briceño-Arias, L.M. & Combettes, P.L. A strongly convergent primal–dual method for nonoverlapping domain decomposition. Numer. Math. 133, 443–470 (2016). https://doi.org/10.1007/s00211-015-0751-4

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  • DOI: https://doi.org/10.1007/s00211-015-0751-4

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