Page 508 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 508
MULTISCALE MODELING AND METHODS: APPLICATION IN ENGINEERING,
BIOLOGY AND MEDICINE (MS-80)

On weakly singular kernels arising in equations set on a graph, modelling
a flow in a networtk of thin tubes

Éric Canon, eric.canon@univ-st-etienne.fr
UJM Saint-Étienne, Institut Camille Jordan, CNRS UMR 5208, France

Coauthors: Frédéric Chardard, Grigory Panasenko, Olga Štikoniene˙

This talk follows the talk by Frédéric Chardard entitled: Numerical solution of the viscous
flows in a network of thin tubes: equations on the graph. These equations are set on a 1D graph
and were obtained by letting the diameters of the tubes tend to zero in some asymptotic process.
They are characterized by a convolution in time in the diffusion operator, with a weakly singular
kernel in time that are computed from the solution of local heat equations (with Dirichlet Con-
ditions) in 2D domains that represent the cross sections of the tubes of the initial network. This
talk is more about these kernels: theoretical results and numerical computations of the kernels.
We obtain in particular asymptotic expansions for small times in different ways: for smooth
cross section inspired by techniques developped by Gie, Hamouda, Jung and Temam (Singular
Perturbations and Boundary Layers, Springer), or for specific cross sections (rectangles, disks,
equilateral triangles) with specific techniques.

Homogenization for elliptic operators in a strip perforated along a curve

Giuseppe Cardone, gcardone@unisannio.it
Università del Sannio, Italy

We consider an elliptic operator in a planar infinite strip perforated by small holes along a curve:
we impose mixed classical boundary conditions (Dirichlet, Neumann and Robin) on the holes,
assuming that the perforation is non-periodic and satisfies rather weak assumptions.

We describe the homogenized operators, establish the norm resolvent convergence of the
perturbed resolvents to the homogenized one, prove the estimates for the rate of convergence
and study the convergence of the spectrum.

Based on a joint work with D. Borisov and T. Durante

Numerical solution of the viscous flows in a network of thin tubes:
equations on the graph

Frédéric Chardard, frederic.chardard@univ-st-etienne.fr
UJM Saint-Étienne, Institut Camille Jordan (UMR 5208), France
Coauthors: Éric Canon, Olga Štikoniene˙, Grigory Panasenko

A non-stationary flow in a network of thin tubes is considered. Its one-dimensional approx-
imation was proposed in a paper by G.Panasenko and K.Pileckas [1]. It consists of a set of
equations with weakly singular kernels, on a graph, for the macroscopic pressure. A new dif-
ference scheme for this problem is proposed. Several variants are discussed. Stability and
convergence are studied theoretically and numerically. Numerical results are compared to the
direct numerical solution of the full dimension Navier-Stokes equations.

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