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7tiscale Finite Element Method, where we construct a multiscale space using solution of the local spectral prob-
lems on the snapshot space related to the coarse grid edges. We present numerical results for the Helmholtz
problem in perforated domain with Dirichlet boundary condition on perforations. Proposed method are stud-
ied for a different wave numbers and numbers of the edge multiscale basis functions.
Embedded discrete fracture model on structured grids
D. Y. Nikiforov
M. K. Ammosov North-Eastern Federal University, Yakutsk
Email: dju92@mail.com
DOI 10.24412/cl-35065-2021-1-02-82
An approximation of the embedded discrete fracture model EDFM by the finite element method is con-
sidered. The paper proposes to use exponential functions instead of the Dirac delta function [1]. With this ap-
proach, instead of a separate computational mesh for fractures, a mesh for a porous medium can be used. The
results of numerical experiments demonstrate the efficiency of the proposed approach.
This work was supported by the Ministry of science and higher education of the Russian Federation, supplementary
agreement N075-02-2020-1542/1, April 29, 2020.
References
1. Nikiforov D. Y., Stepanov S. P. Numerical simulation of the embedded discrete fractures by the finite element
method //J. of Physics: Conference Series. � IOP Publishing, 2019. Vol. 1158. No. 3. P. 032038.
Multiscale finite element technique for mathematical modelling of multi-physics processes
in the near-wellbore region
E. P. Shurina1,2, N. B. Itkina1,3, D. A. Arhipov1,2, D. V. Dobrolubova1,2, A. Yu. Kutishcheva1,2, S. I. Markov1,2,
N. V. Shtabel1,2, E. I. Shtanko1,2
1The Trofimuk Institute of Petroleum Geology and Geophysics SB RAS
2Novosibirsk State Technical University
3Institute of Computational Technologies SBRAS
Email: shurina@online.sinor.ru
DOI 10.24412/cl-35065-2021-1-00-56
In borehole physic, the results of the direct mathematical modelling of multi-physical phenomena are
used to control drilling and well operation. Electromagnetic and acoustic measurements are the most accessi-
ble indirect methods for determining the thermal, transport and mechanical properties of rock samples in the
near-wellbore zone. Mathematical modelling is one of the technologies used for solving multi-physical prob-
lems. A multi-physical problem is formulated as a system of partial differential equations with special interface
conditions coupling mathematical models of physical processes. The near-wellbore region is characterized by a
multi-scale geometric structure, high contrast and anisotropy of physical parameters. The discretization meth-
od should take into account the specifics of the problem and preserve the global regularity of mathematical
models at a discrete level. The paper presents modified variational formulations of multiscale non-conforming
finite element methods for mathematical modelling of electromagnetic and acoustic fields in fluid-saturated
media at various temperatures and mechanical loads. To solve the discretized mathematical models, special
multilevel solvers are developed. The results of three-dimensional mathematical modelling using model rock
samples from the near-wellbore zone are presented.
The research was supported by RSF Project No. 20-71-00134 (coupled heat and mass transfer), Project No. 0266-
2019-0007 (hydrodynamics and acoustics), Project No. 0331-2019-0015 (electromagnetism).
Analysis of ionospheric irregularities based on multi-instrumental data
D. Sidorov1,2, Yu. Yasukevuch1, E. Astafyeva3, A. Garashenko1, A. Yasyukevich1, A. Oinats1, A. Vesnin1
1Institute of Solar-Terrestrial Physics SB RAS
2Institute of Energy Systems SB RAS
3Universite de Paris, Institut de Physique du Globe de Paris, CNRS UMR 7154, France
Email: contact.dns@gmail.com
DOI 10.24412/cl-35065-2021-1-03-06
The most complex ionospheric phenomena occur in the areas of auroral ovals. These areas are character-
ised by intense small-scale ionospheric inhomogeneities that exist in both calm and disturbed geomagnetic
conditions. Such irregularities could result in radio wave scattering, GNSS (global navigation satellite system)
positioning quality deterioration, failures in radio communication, etc. GNSS ROTI (rate of total electron con-
tent index) datasets along with other datasets are available to study complex dynamics of ionospheric irregu-
larities. This report analyses the auroral oval dynamics datasets, based on GNSS global network, coherent ra-
dars data, and satellite data. The SIMuRG system (https://simurg.iszf.irk.ru/) is employed. The auroral oval re-
gions corresponds to high values of ROTI, therefore it is possible to separate their location from mid-latitude
data. Coherent scatter radars record signal scattering from the oval boundary. The SuperDARN-like radars lo-
cated in Russia were employed. Satellite data shows sharp variations in field-aligned currents. During magnetic
storms the oval expands equatorward, and small-scale irregularities shifts to mid-latitudes. All the data show
close positions of the oval boundary. The latter makes it possible to use the datasets of different modalities to
estimate the oval boundary. Some advance was achieved by computer vision techniques to find the auroral
oval boundary in the Northern hemisphere. The techniques implemented mathematical morphology to ex-
pand data and decrease data gaps, Otsu techniques and K-means to cluster image data.
This work was supported by the Russian Science Foundation (project RSF 17-77-20005).
Mixed generalized multiscale finite element method for flow problem in thin domains
D. A. Spiridonov1, M. V. Vasilyeva1,2, Ya. Efendiev3, E. Chung4, M. Wang 5
1M. K. Ammosov North-Eastern Federal University, Yakutsk
2Institute for Scientific Computation, Texas A&M University
3Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College
Station, Texas, USA
4Department of Mathematics, The Chinese University of Hong Kong (CUHK), Hong Kong SAR
5Duke University (Durham), USA
Email: d.stalnov@mail.ru
DOI 10.24412/cl-35065-2021-1-02-83
In this work, we consider the construction of the Mixed Generalized Multiscale Finite Element Method
approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the ellip-
tic equation with homogeneous boundary conditions on the domain walls. For reference solution of the prob-
lem, we use a Mixed Finite Element Method on a fine grid that resolves complex geometry on the grid level. To
construct a lower dimensional model, we use the Mixed Generalized Multiscale Finite Element Method, where
we present the construction of the multiscale basis functions for velocity fields. The construction is based on
