Vol 43, No 2 (2019)

For stationary diffusion-type equations, we study the convergence of grid inhomogeneous boundary-value problems of a version of the mimetic finite difference (MFD) technique in which grid scalars are defined inside grid cells and grid vectors are specified by their local normal coordinates on the plane faces of grid cells. Grid equations and boundary conditions are formulated in operator form using consistent grid analogs of invariant first-order differential operators and of boundary operators. Convergence is studied on the basis of the theory of operator difference schemes; i.e., a priori estimates for the norm of the solution error in terms of the norm of the approximation error are obtained that guarantee convergence of the first order under inhomogeneous boundary conditions of the first, second, and third kind in a domain with a curvilinear boundary. Grid analogs of embedding inequalities and approximation relations obtained earlier are used.

The problem of constructing an explicit formula for a majorant of a number sequence specified by recurrence relations is considered. Problems of this kind arise when estimating the error of recursive methods for calculating certain functions of a real variable. A special approach to studying the given recurrence relations is proposed in this work, based on which it is established that a majorant can be provided by a quite specific formula and belongs to the class of subexponential functions.

A graph is called pseudocubic if the degrees of all its vertices, with a single exception, do not exceed three, and the degree of an exceptional vertex does not exceed four. In this work, it is proved that the vertices of a pseudocubic graph without induced subgraphs that are isomorphic to K₄ or K₄⁻ can be colored in three colors. In addition, it is shown that the problem of 3-coloring of pseudocubic graphs can be solved using a polynomial algorithm.