Home – Home – Jornals – Numerical Analysis and Applications 2022 number 4

The present paper is addressed to the estimation of the local (point-wise) 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 ill-posed 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 two-dimensional 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.

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.

This article discusses the mixed finite element method combined with backward-Euler method to study the hyperbolic p-bi-Laplace equation, where the existence and uniqueness of solution for discretized problem is shown in Lebesgue Sobolev spaces. The mixed formulation and the inf-sup 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.

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.

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.

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 well-known ν-convergence used in the case of classical spectrum approximation. This new vision allows us to see the ν-convergence 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.

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 ndelin-Lobachevsky-Graeffe method for squaring the roots. Numerical results on the construction of stability domains of Ams-Bashforth methods of order 3-11 are given.

An experimental study of the efficiency of 3D bounry value problem solvers on the regular subgrids of quasi-structured parallelepipel grids has been carried out. Five solvers are considered: three iterative: the successive over-relaxation 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 over-relaxation method.

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.