Multiscale methods based on coupled partial differential equations defined on bulk and embedded manifolds are still poorly explored from the theoretical standpoint, although they are successfully used in applications, such as microcirculation and flow in perforated subsurface reservoirs. This work aims at shedding light on some theoretical aspects of a multiscale method consisting of coupled partial differential equations defined on one-dimensional domains embedded into three-dimensional ones. Mathematical issues arise because the dimensionality gap between the bulk and the inclusions is larger than one, that is the high dimensionality gap case. First, we show that such model derives from a system of fully three-dimensional equations, by the application of a topological model reduction approach. Secondly, we rigorously analyze the problem, showing that the averaging operators applied for the model reduction introduce a regularization effect that resolves the issues due to the singularity of solutions and to the ill-posedness of restriction operators. Then, we exploit the structure of the model reduction technique to analyze the modeling error. This study confirms that for infinitesimally small inclusions, the modeling error vanishes. Finally, we discretize the problem by means of the finite element method and we analyze the approximation and the model error by means of numerical experiments.
Mots-clés : Embedded geometric multiscale, heterogeneous domain dimensionality, model error analysis
@article{M2AN_2019__53_6_2047_0,
author = {Laurino, Federica and Zunino, Paolo},
title = {Derivation and analysis of coupled {PDEs} on manifolds with high dimensionality gap arising from topological model reduction},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2047--2080},
publisher = {EDP-Sciences},
volume = {53},
number = {6},
year = {2019},
doi = {10.1051/m2an/2019042},
mrnumber = {4040715},
zbl = {1435.35155},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2019042/}
}
TY - JOUR AU - Laurino, Federica AU - Zunino, Paolo TI - Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 2047 EP - 2080 VL - 53 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019042/ DO - 10.1051/m2an/2019042 LA - en ID - M2AN_2019__53_6_2047_0 ER -
%0 Journal Article %A Laurino, Federica %A Zunino, Paolo %T Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 2047-2080 %V 53 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019042/ %R 10.1051/m2an/2019042 %G en %F M2AN_2019__53_6_2047_0
Laurino, Federica; Zunino, Paolo. Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 2047-2080. doi : 10.1051/m2an/2019042. http://www.numdam.org/articles/10.1051/m2an/2019042/
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