Analysis of patch substructuring methods

Martin Gander; Laurence Halpern; Frédéric Magoulès; Francois Roux

International Journal of Applied Mathematics and Computer Science (2007)

  • Volume: 17, Issue: 3, page 395-402
  • ISSN: 1641-876X

Abstract

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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 convergencerate than both the algebraic and the geometric one. We complement our results by numerical experiments.

How to cite

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Gander, Martin, et al. "Analysis of patch substructuring methods." International Journal of Applied Mathematics and Computer Science 17.3 (2007): 395-402. <http://eudml.org/doc/207845>.

@article{Gander2007,
abstract = {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 convergencerate than both the algebraic and the geometric one. We complement our results by numerical experiments.},
author = {Gander, Martin, Halpern, Laurence, Magoulès, Frédéric, Roux, Francois},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {patch substructuring methods; Schur complement methods; Schwarz domain decomposition methods; optimized Schwarz methods; finite element method; finite difference method; Neumann transmission conditions; convergence; numerical experiments},
language = {eng},
number = {3},
pages = {395-402},
title = {Analysis of patch substructuring methods},
url = {http://eudml.org/doc/207845},
volume = {17},
year = {2007},
}

TY - JOUR
AU - Gander, Martin
AU - Halpern, Laurence
AU - Magoulès, Frédéric
AU - Roux, Francois
TI - Analysis of patch substructuring methods
JO - International Journal of Applied Mathematics and Computer Science
PY - 2007
VL - 17
IS - 3
SP - 395
EP - 402
AB - 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 convergencerate than both the algebraic and the geometric one. We complement our results by numerical experiments.
LA - eng
KW - patch substructuring methods; Schur complement methods; Schwarz domain decomposition methods; optimized Schwarz methods; finite element method; finite difference method; Neumann transmission conditions; convergence; numerical experiments
UR - http://eudml.org/doc/207845
ER -

References

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  10. Magoulès F., Roux F. -X. and Series L. (2006): Algebraic approximation of Dirichlet-to-Neumann maps for the equations of linear elasticy. Computer Methods in Applied Mechanics and Engineering, Vol. 195, No. 29-32, pp. 3742-3759. 
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