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Error of an eFDDM: What Do Matched Asymptotic Expansions Teach Us?

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Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 104)

Abstract

In this paper, we analyze the approximation error of an explicit Fuzzy Domain Decomposition Method (eFDDM) (Gander and Michaud, Fuzzy domain decomposition: a new perspective on heterogeneous DD methods, in Domain Decomposition Methods in Science and Engineering XXI, ed. by J. Erhel, M.J. Gander, L. Halpern, G. Pichot, T. Sassi, O.B. Widlund. Lecture Notes in Computational Science and Engineering. Springer, Berlin, 2013) using matched asymptotic expansions (Cousteix and Mauss, Asymptotic Analysis and Boundary Layers, Springer, Berlin, 2007). We show that the global convergence of the method for an advection dominated diffusion problem is of order \(\mathcal{O}(\nu )\) and have numerical evidence that the method is of order \(\mathcal{O}(\nu ^{3/2})\) in the boundary layer. Our results generalize the results of Gander and Martin (An asymptotic approach to compare coupling mechanisms for different partial differential equations, in Domain Decomposition Methods in Science and Engineering XX, ed. by R. Bank, M. Holst, O.B. Widlund, J. Xu. Lecture Notes in Computational Science and Engineering, Springer, Berlin, 2012) to this new method and show that the eFDDM is a viable alternative to other coupling methods.

Keywords

  • Boundary Layer
  • Approximation Error
  • Membership Function
  • Domain Decomposition
  • Coupling Method

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Fig. 1

Notes

  1. 1.

    This is only a technicality to guaranty the wellposedness of the trace h 1∕2 u on ∂ Ω. Typical smooth “plateau” functions satisfy this condition.

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Correspondence to Jérôme Michaud .

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Michaud, J., Cocquet, PH. (2016). Error of an eFDDM: What Do Matched Asymptotic Expansions Teach Us?. In: Dickopf, T., Gander, M., Halpern, L., Krause, R., Pavarino, L. (eds) Domain Decomposition Methods in Science and Engineering XXII. Lecture Notes in Computational Science and Engineering, vol 104. Springer, Cham. https://doi.org/10.1007/978-3-319-18827-0_33

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