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6References
1. Cerna D. Cubic spline wavelets with four vanishing moments on the interval and their applications to option
pricing under Kou mode // International J. of wavelets, multiresolution and information processing. 2019. V. 17, N. 1.
Article N. 1850061.
2. Koro K., Abe K. Non-orthogonal spline wavelets for boundary element analysis // Engineering analysis with
boundary elements. 2001. V. 25. P. 149-164.
3. Shumilov B. M. Algorithm with splitting for cubic spline wavelets with two zero moments on an interval // Siberian
electronic mathematical reports. 2020. V. 17. P. 2105-2121.
4. Pissanetzky S. Sparse matrix technology. London: Academic press, 1984.
5. Shumilov B. M. Semi-orthogonal spline-wavelets with derivatives and the algorithm with splitting // Numerical
analysis and applications. 2017. V. 10, N. 1. P. 90-100.
On seven-diagonals splitting for cubic spline wavelets with six vanishing moments on an interval
B. M. Shumilov
Tomsk State University of Architecture and Building
Email: sbm@tsuab.ru
DOI 10.24412/cl-35065-2021-1-00-55
In this study, we use a zeroing property of the first six moments for constructing a splitting algorithm for
cubic spline wavelets. First, we construct a new system of cubic basic spline-wavelets, realizing orthogonal
conditions to all polynomials up to fifth degrees [1]. Then, using the homogeneous Dirichlet boundary condi-
tions [2], we adapt spaces to the closed interval. The originality of the study consists of obtaining implicit rela-
tions connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and
wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate
matrix. After excluding the even rows of the system, in contrast to the case with two zero moments [3], the
resulting transformation matrix has five or seven (instead of three) diagonals. For a seven-diagonal matrix, the
presence of a strict diagonal dominance over the columns [4] is proved. The comparative results of numerical
experiments on approximating and calculating the derivatives of a discrete function are presented.
References
1. Koro K., Abe K. Non-orthogonal spline wavelets for boundary element analysis // Engineering Analysis with
Boundary Elements. 2001. V. 25. P. 149-164.
2. Cerna D. Cubic spline wavelets with four vanishing moments on the interval and their applications to option
pricing under Kou mode // International J. of Wavelets, Multiresolution and Information Processing. 2019. V. 17, N. 1.
Article N. 1850061.
3. Shumilov B. Algorithm with splitting for cubic spline wavelets with two zero moments on an interval // Siberian
Electronic Mathematical Reports. 2020. V. 17. P. 2105-2121.
4. Pissanetzky S. Sparse Matrix Technology. London: Academic Press, 1984.
Hierarchical basis on tetrahedra for mixed finite element formulation of the Darcy problem
E. P. Shurina1,2, N. B. Itkina2, S. A. Trofimova1,2
1Trofimuk Institute of Petroleum Geology and Geophysics SB RAS
2Novosibirsk State Technical University
Email: svetik-missy@mail.ru, TrofimovaSA@ipgg.sbras.ru, itkina.nat@yandex.ru, shurina@online.sinor.ru
DOI 10.24412/cl-35065-2021-1-00-57
The solution of a certain class of applied problems in the oil industry involves the use of mathematical
models that describe complex processes associated with the intensification and development of hydrocarbon
fields. Mixed variational formulation turn out to be effective for determining the explicit behavior of the nor-
mal velocity component at the boundary of the modeling domain, however, they involve finding a solution in
two spaces [1]. In this paper the problem of constructing a specialized hierarchical basis systems on tetrahe-
dral finite elements in the H1-space for pressure and in the Hdiv-space for velocity is investigated and also the
influence of this basis on the properties of a matrix of a discrete analogue of a non-conformal mixed formula-
tion based on the discontinuous Galerkin method is studied [2].
This work was supported by the Project No. 0266-2019-0007.
References
1. F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Masud. Mixed discontinuous Galerkin methods for Darcy flow // J. of
Scientific Computing. 2005. V. 22, No. 1. P. 119-225.
2. P. Solin, K. Segeth, I. Dolezel. High-order finite element methods, ChapmananHall, CRC, 2004. 388 p.
Particle motions for the gas dynamics equations with the special state equation
D. T. Siraeva1
1Mavlyutov Institute of Mechanics UFRC RAS
Email: sirdilara@gmail.com
DOI 10.24412/cl-35065-2021-1-00-59
The gas dynamics equations with the state equation of special form are considered. The state equation is
a pressure equal to the sum of two functions. The first function depends on density, and the second function
depends on entropy [1]. The system admits a 12-dimensional Lie algebra. An optimal system of dissimilar sub-
algebras of the Lie algebra was constructed in [2]. Invariant submodels are calculated for 2- and 3-dimensional
subalgebras. Exact solutions were found for some submodels.
The motion of particles and volumes according to the exact solutions is considered due to using the com-
puter mathematics system Maple.
The work was supported by the Russian Foundation for Basic Research (project no. 18-29-10071) and partially from
the Federal Budget by the State Target (project no. 0246-2019-0052).
References
1. Ovsyannikov L.V. The �podmodeli� program. Gas dynamics // J. of Appl. Math. and Mechan. 1994. V. 58, N. 4.
P. 601-627. Doi:10.1016/0021-8928(94)90137-6.
2. Siraeva D.T. Optimal system of non-similar subalgebras of sum of two ideals // Ufa Mathematical J.. 2014. V. 6,
N. 1. P. 90-103. Doi:10.13108/2014-6-1-90.
The coupling of the vectorial and scalar boundary element methods
S. A. Sivak, M. E. Royak, I. M. Stupakov
Novosibirsk State Technical University
Email: siwakserg@yandex.ru
DOI 10.24412/cl-35065-2021-1-00-60
The vectorial boundary element method is a tool applied to solve electromagnetic problems in a media
with consideration of eddy currents [1]. It�s also known as the boundary element method for eddy current
problems [2, 3]. The use of this method brings certain difficulties, one of which is the problem of zero wave
number in the subdomains adjacent to the domain where the eddy currents should be considered. As a means
to mitigate the computational difficulty, we present in this paper the coupling with the scalar potential. The
