Analysis of patch substructuring methods

• Autorzy: Gander M. J. , Halpern L. , Magoules F. , Roux F. X.
• Języki publikacji: EN

Abstrakty

EN

Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains, condensated on the interfaces, to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one patch per interface. We show that this method is equivalent to an overlapping Schwarz method using Neumann transmission conditions. This equivalence is obtained by first studying a new, algebraic patch method, which is equivalent to the classical Schwarz method with Dirichlet transmission conditions and an overlap corresponding to the size of the patches. Our results motivate a new method, the Robin patch method, which is a linear combination of the algebraic and the geometric one, and can be interpreted as an optimized Schwarz method with Robin transmission conditions. This new method has a significantly faster convergence rate than both the algebraic and the geometric one. We complement our results by numerical experiments.

Słowa kluczowe

EN

Schwarz domain decomposition methods; Schur complement methods; patch substructuring methods; optimized Schwarz methods

PL

metoda rozkładu; metoda dopełniacza Schura; zoptymalizowana metoda Schwarza

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2007
• Tom: Vol. 17, no 3
• Strony: 395--402
• Opis fizyczny: Bibliogr. 15 poz., rys., tab., wykr.

Twórcy

autor

Gander M. J.

• Section de Mathématiques, Université de Genève, 1211 Genève, Switzerland

autor

Halpern L.

• LAGA, Institut Galilée, Université Paris 13, 93430 Villetaneuse, France

autor

Magoules F.

• Applied Mathematics and Systems Laboratory, Ecole Centrale Paris, 92295 Châtenay-Malabry Cedex, France

autor

Roux F. X.

• High Performance Computing Research Unit, ONERA, 92322 Chatillon, France

Bibliografia

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• [3] Japhet C. (1998): Optimized Krylov-Ventcell method. Application to convection-diffusion problems. Proceedings of the 9th International Conference Domain Decomposition Methods, Bergen, Norway, pp. 382-389.
• [4] Le Tallec P. (1994): Domain decomposition methods in computational mechanics, In: Computational Mechanics Advances, (J. Tinsley Oden, Ed.). North-Holland, Amsterdam, Vol. 1, No. 2, pp. 121-220.
• [5] Lions P.-L. (1988): On the Schwarz alternating method. I. Proceedings of the 1st International Symposium Domain Decomposition Methods for Partial Differential Equations, Philadelphia, PA: SIAM, pp. 1-42.
• [6] Lions P.-L. (1990): On the Schwarz alternating method. III: A variant for nonoverlapping subdomains. Proceedings of the 3rd International Symposium Domain Decomposition Methods for Partial Differential Equations, Philadelphia, PA: SIAM, pp. 202-223.
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• [8] Magoulès F., Roux F.-X. and Salmon S. (2004b): Optimal discrete transmission conditions for a non-overlapping domain decomposition method for the Helmholtz equation. SIAM Journal on Scientific Computing, Vol. 25, No. 5, pp. 1497-1515.
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