Volume 34, Issue 3

A system of ordinary differential equations describing a stationary distribution of a Markov process with the phase space R × {1, 2, …, M } is considered. The finite element method is proposed for calculation of its solution as a probability density.

The paper is focused on justification of a statistical modelling algorithm for solution of the nonlinear kinetic Boltzmann equation on the base of a projection method. Hermite functions are used as an orthonormal basis. The error of approximation of a function by a partial sum of Hermite functions series is estimated in the L 2 norm. The estimates are compared for two variants of the projection method in the case of solutions to the homogeneous gas relaxation problem with a known solution.

Effective coefficients for convective diffusion equations are obtained. Correlated random fields of conductivity and porosity are approximated by multiplicative cascades with log-normal distributions of probabilities. The theoretical result is checked numerically for the case of 3D plane flow of an incompressible fluid.

In this paper we present a new solution method for underdetermined systems of nonlinear equations in a neighborhood of a certain point of the variety of solutions where the Jacoby matrix has incomplete rank. Such systems are usually called degenerate. It is known that the Gauss–Newton method can be used in the degenerate case. However, the variety of solutions in a neighborhood of the considered point can have several branches in the degenerate case. Therefore, the analysis of convergence of the method requires special techniques based on the constructions of the theory of p -regularity and p -factor-operators.

Solving linear divergence-curl system with Dirichlet conditions is reduced to finding an unknown vector function in the space of piecewise-polynomial gradients of harmonic functions. In this approach one can use the boundary least squares method with a harmonic basis of a high order of approximation formulated by the authors previously. The justification of this method is given. The properties of the bilinear form and approximating properties of the basis are investigated. Convergence of approximate solutions is proved. A numerical example with estimates of experimental orders of convergence in Vhp$\begin{array}{} {\bf V}_h^p \end{array}$-norm for different parameters h , p ( p ⩽ 10) is presented. The method does not require specification of penalty weight function.