In this paper, we develop in a general framework a non overlapping Domain Decomposition Method that is proven to be well-posed and converges exponentially fast, provided that specific transmission operators are used. These operators are necessarily non local and we provide a class of such operators in the form of integral operators. To reduce the numerical cost of these integral operators, we show that a truncation process can be applied that preserves all the properties leading to an exponentially fast convergent method. A modal analysis is performed on a separable geometry to illustrate the theoretical properties of the method and we exhibit an optimization process to further reduce the convergence rate of the algorithm.
Keywords: Domain decomposition methods, exponentially fast convergent methods, integral operators, norms of fractional order Sobolev spaces, pseudo-differential operators
@article{M2AN_2020__54_3_775_0,
author = {Collino, Francis and Joly, Patrick and Lecouvez, Matthieu},
title = {Exponentially convergent non overlapping domain decomposition methods for the {Helmholtz} equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {775--810},
year = {2020},
publisher = {EDP Sciences},
volume = {54},
number = {3},
doi = {10.1051/m2an/2019050},
mrnumber = {4080785},
zbl = {1437.65219},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2019050/}
}
TY - JOUR AU - Collino, Francis AU - Joly, Patrick AU - Lecouvez, Matthieu TI - Exponentially convergent non overlapping domain decomposition methods for the Helmholtz equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 775 EP - 810 VL - 54 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2019050/ DO - 10.1051/m2an/2019050 LA - en ID - M2AN_2020__54_3_775_0 ER -
%0 Journal Article %A Collino, Francis %A Joly, Patrick %A Lecouvez, Matthieu %T Exponentially convergent non overlapping domain decomposition methods for the Helmholtz equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 775-810 %V 54 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2019050/ %R 10.1051/m2an/2019050 %G en %F M2AN_2020__54_3_775_0
Collino, Francis; Joly, Patrick; Lecouvez, Matthieu. Exponentially convergent non overlapping domain decomposition methods for the Helmholtz equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 775-810. doi: 10.1051/m2an/2019050
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