Abstract
Optimized Schwarz methods have attracted tremendous attention because of their excellent performance. For circular domain decompositions, Gander and Xu (SIAM J. Numer. Anal. 52(4): 1981–2004, 2014; Math. Comput., 2016b) showed that the interface curvature enters the optimized transmission condition and the corresponding convergence rate estimate. Since where the curvature is the only interface characteristic, how the curved interface affects the performance of the optimized Schwarz method is still not completely clear. In this short paper, we show, for an example of domain decomposition with parabolic interfaces, that not only the interface curvature, but also other characteristics of the interfaces will enter as well the optimized transmission parameter. We use numerical experiments to illustrate our theoretical findings.
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Notes
- 1.
The length of the interface σ = σ 0 is easy to calculate to be \(\frac{\sigma _{0}^{2}} {2} \mathtt{arcsinh}(\frac{1} {\sigma _{0}} ) + \frac{1} {2}\sqrt{\sigma _{0 }^{2 } + 1}\).
References
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Acknowledgements
The author “Y. Xu” was partly supported by NSFC-11671074, 11471047, CPSF-2012M520657 and the Science and Technology Development Planning of Jilin Province 20140520058JH.
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Gander, M.J., Xu, Y. (2017). Optimized Schwarz Methods for Domain Decompositions with Parabolic Interfaces. In: Lee, CO., et al. Domain Decomposition Methods in Science and Engineering XXIII. Lecture Notes in Computational Science and Engineering, vol 116. Springer, Cham. https://doi.org/10.1007/978-3-319-52389-7_33
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DOI: https://doi.org/10.1007/978-3-319-52389-7_33
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