## Uniform in ε convergence of the standard Galerkin finite element method in L2-norm for the singularly-perturbed elliptic problems on a priori chosen meshes

Convection-diffusion problems are models for describing the transportation of matters in a diffusive medium. They are also a part of more general equations, such as the Navier-Stokes equations appearing in fluid flow problems. We consider the linear convection-diffusion problem of the special form and show that under the corresponding conditions...

## Application of Monte Carlo method to determination of telegraph equation coefficients

In this paper, a probabilistic approach to solving some inverse and direct problems to the telegraph equation is presented. The multidimensional cases and specific features of the inverse problems, where it is commonly required to determine only the functional of solution, make the application of Monte Carlo method reasonable.

## Study of radiation transfer using the computational educational manual on Monte Carlo methods

It is well-known that Monte Carlo methods are used for solving various urgent problems in mathematical physics. A lot of new numerical stochastic algorithms and models were elaborated in Institute of Computational Mathematics and Mathematical Geophysics (ICM and MG) SD RAS during last years. The Monte Carlo schemes can be...

## On a shape preserving interpolation by local VP-splines

The problem we discuss here is how to construct the local smooth *VP*-splines preserving the monotonicity of data from the point of view of the regions of their parameters. Also, we will give a practical recommendation how to satisfy these conditions if data and the slopes are given and fixed.

## Algorithms for the construction of quasiregular hierarchical grids

The problem of the construction of "good" grids for the numerical solution of multidimensional boundary value problems (BVPs) in complicated computational domains has a big history and extensive special literature.

There are two main conventional requirements for the discretization of BVPs. The first one consists in the approximation quality which...

## Fictitious domain method for fourth-order elliptic problem

In this paper, we study the convergence of the fictitious domain method for solving a system of grid equations for the finite element method that approximates the third boundary problem for the differential equation
Δ2*u* + *au* = *f*
in the piecewise bicubic Hermit interpolations subspace of *W...*

## Multivariant partial solution of linear systems of 5-point equations by the diagonal transfer method

Using the diagonal transfer method, a partial solution of a linear system of equations is found. The description of corresponding algorithm is given. Computational costs of the algorithm are discussed.

## Pseudo-orthogonal polynomials

The classical problem of the transformation of the orthogonality weights of polynomials by theory of the space *Rn* is discussed. The described system of polynomials - pseudo-orthogonal on the discrete set of *n* points - is a new result. The polynomials of this system, as the orthogonal ones, are...

## An optimal multilevel method for computing the smallest eigenpair

In this paper, we are interested in computing the smallest eigenvalue and its corresponding eigenvector of a large symmetric positive definite matrix.

## Finite element trace theorems

The theorems on traces of functions from the Sobolev spaces play an important role in studying boundary value problems of mathematical physics. These theorems are commonly used for a priori estimates of the stability with respect to boundary conditions. The trace theorems play also very important role to construct and...

## Direct simulation Monte Carlo method for stationary nonlinear Boltzmann equation

A new version of the direct Monte Carlo method for solving boundary value problems for the Boltzmann equation is presented. In contrast to the conventional approach, we do not solve the problem via stabilization in time; when evaluating functionals of the solution to the Boltzmann equation, the random trajectories are...

## Dissipativity of generalized Maxwell-Leontovich operator

This paper completes the analysis of the Maxwell operator with impedance boundary conditions for arbitrary time dependence.

## Fast coding-decoding algorithm in fractal image compression via spherical ranges classification

The paper deals with the new classification of the ranges in image based on simple geometrical consideration. This approach leads to the fast algorithms for the training of fractal bases and fast coding-decoding processes in the image compression. Some numerical examples are also presented.

## Optimization of energy functional in abstract spline interpolation: criteria and some results for splines with tension

The aim of this paper is to introduce some formal procedures for the optimization of parameters in energy functional for variational spline interpolation. For the abstract splines with the tension we obtain the representation formula which shows the structure of dependence of spline with the tension with respect to tension...

## Properties of matrices in methods of constructing an interpolating spline via the coordinates of its derivatives in B-spline basis

We consider an interpolation problem. We have the given values
*fi* = *f*(*xi*)
of some periodic function *f*(*x*)
of the period *b - a* in the nodes of a mesh

Δ: *a* = *x*0 < *x*1 < ... < *x*N = *b*.

It is required to construct a (*b - a*)-periodic interpolating spline *S*(*x*) of the degree 2*n* - 1 (*N* ≥ 2*n*).