

2022 year, number 4
A.K. Alekseev, A.E. Bondarev
Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, Russia
Keywords: pointwise approximation error, ensemble of numerical solutions, Richardson extrapolation, Inverse problem, Euler equations
Abstract >>
The present paper is addressed to the estimation of the local (pointwise) approximation error on the ensemble of the numerical solutions obtained using independent algorithms. The variational inverse problem is posed for th approximation error estimation. The considered problem is illposed due to invariance of the governing equations to the shift transformations. By this reason, the zero order Tikhonov regularization is applied. The numerical tests for the twodimensional equations describing the inviscid compressible flow are performed in order to verify the efficiency of considered algorithm. The estimates of approximation errors, obtained by the considered inverse problem, demonstrate the satisfactory accornce with the Richardson extrapolation results at significantly less computational costs.

S.A. Gusev^{1,2}
^{1}Institute Computational Mathematics and Mathematical Geophysics Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia ^{2}Novosibirsk State Technical University, Novosibirsk, Russia
Keywords: diffusion process, variance of the Monte Carlo method estimation, stochastic differential equations, reflecting bounry, Euler method
Abstract >>
The estimation of the functional of the diffusion process in a domain with a reflecting bounry, which is obtained on the basis of numerical modeling of its trajectories, is considered. The value of this functional coincides with the solution at a given point of a bounry value problem of the third kind for a parabolic equation. A formula is obtained for the limiting value of the variance of this estimate under decreasing step in the Euler method. To reduce the variance of the estimate, a transformation of the bounry value problem is used, similar to the one that was previously proposed in the case of an absorbing bounry.

Manal Djaghout^{1}, Abderrazak Chaoui^{1}, Khaled Zennir^{2}
^{1}UniversitÃ¨ 8 Mai 1945 Guelma, Guelma, AlgÃ¨rie ^{2}Qassim University, ArRass, Saudi Arabia
Keywords: evolution pbiLaplace equation, mixed finite element method, infsup condition and mixed formulation, existence and uniqueness
Abstract >>
This article discusses the mixed finite element method combined with backwardEuler method to study the hyperbolic pbiLaplace equation, where the existence and uniqueness of solution for discretized problem is shown in Lebesgue Sobolev spaces. The mixed formulation and the infsup condition are then given to prove the well posed of the scheme and the optimal a priori error estimates for fully discrete schemes is extracted. Finally, a numerical example is given to confirm the theoretical results obtained.

M.I. Ivanov, I.A. Kremer, Yu.M. Laevsky
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
Keywords: Neumann problem, generalized formulation, Lagrange multipliers, mixed finite element method, saddle point algebraic linear system, matrix kernel
Abstract >>
This paper proposes a new method for the numerical solution of a pure Neumann problem for the diffusion equation in a mixed formulation. The method is based on the inclusion of a condition of unique solvability of the problem in one of the equations of the system with a subsequent decrease in its order by using a Lagrange multiplier. The unique solvability of the problem obtained and its equivalence to the original mixed formulation in a subspace are proved. The problem is approximated on the basis of a mixed finite element method. The unique solvability of the resulting saddle system of linear algebraic equations is investigated. Theoretical results are illustrated by computational experiments.

Saidkhakim Ikramov^{1}, Ali Mohammad Nazari^{2}
^{1}Lomonosov Moscow State University, Moscow, Russia ^{2}University of Arak, Arak, Islamic Republic Iran
Keywords: congruence transformation, unitoid, cosquare, canonical angle, circulant
Abstract >>
A unitoid matrix is a square complex matrix that can be brought to diagonal form by a Hermitian congruence transformation. The canonical angles of a nonsingular unitoid matrix A are (up to the factor 1/2) the arguments of the eigenvalues of the cosquare of A, which is the matrix A^{*}A. We derive an estimate for the derivative of an eigenvalue of the cosquare in the direction of the perturbation in A^{*}A caused by a perturbation in A.

Somia Kamouche, Hamza Guebbai
Laboratoire des MathÃ©matiques AppliquÃ©es et de ModÃ©lisation, UniversiÃ© 8 Mai 1945, Guelma, AlgÃ¨rie
Keywords: generalized spectrum, Î½convergence, property U, spectral approximation
Abstract >>
In this paper, we introduce a new convergence mode to deal with the generalized spectrum approximation of two bounded operators. This new technique is obtained by extending the wellknown νconvergence used in the case of classical spectrum approximation. This new vision allows us to see the νconvergence assumption as a special case of our new method compared to the hypotheses needed in old methods, those required in this paper are weaker. In addition, we prove that the property U holds, which solves the spectral pollution problem arising in spectrum approximation of unbounded operator.

I.V. Kireev^{1,2}, A.E. Novikov^{2}, E.A. Novikov^{1,2}
^{1}Institute of Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russia ^{2}Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk, Russia
Keywords: AmsBashforth method, locus, stability domain, Bernoulli method, ndelinLobachevskyGraeffe method
Abstract >>
A new algorithm is proposed for obtaining stability domains of multistep numerical schemes. The algorithm is based on Bernoulli's algorithm for computing the greatest in magnitude root of a polynomial with complex coefficients and the ndelinLobachevskyGraeffe method for squaring the roots. Numerical results on the construction of stability domains of AmsBashforth methods of order 311 are given.

Il.A. Klimonov^{1}, V.M. Sveshnikov^{2}
^{1}Novosibirsk State University, Novosibirsk, Russia ^{2}Institute of Computational Mathematics and Mathematical Geophysics of Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia
Keywords: regular subgrids of quasistructured grids, bounry value problem solvers, direct methods, iterative methods, experimental research
Abstract >>
An experimental study of the efficiency of 3D bounry value problem solvers on the regular subgrids of quasistructured parallelepipel grids has been carried out. Five solvers are considered: three iterative: the successive overrelaxation method, the implicit alternating direction method, the implicit incomplete factorization method with acceleration by conjugate gradients, as well as two direct methods: PARDISO and HEMHOLTZ  both from the Intel MKL library. The characteristic features of the conducted research are the following: 1) the subgrids contain a small number of nodes; 2) the efficiency is estimated not only for single calculations, but also mainly for a series of calculations, in each of which a large number of repetitions of solving the problem with different bounry conditions on the same same subgrid. On the basis of numerical experiments, the fastest solver under the given conditions was revealed, which turned out to be the method of successive overrelaxation method.

M.Yu. Kokurin, V.V. Klyuchev
Mary State University, YoshkarOla, Russia
Keywords: wave sensing, hyperbolic equation, coefficient inverse problem, integral equation, uniqueness of solution, quadrature method, conjugate gradient method, parallel calculations
Abstract >>
M.M. Lavrentiev's linear integral equation arises as a result of a special transformation of a nonlinear coefficient inverse wave sensing problem. The completeness of the set of products of regular harmonic functions and Newtonian potentials supported by a segment is proved. As a corollary, we establish the uniqueness of the solution to M.M. Lavrentiev's equation and a related inverse problem of wave sensing. We present results of an approximate solution of this equation by using parallelization of calculations.

