In a rectangular domain, we consider the 5-point approximate solution of the multilevel nonlocal boundary value problem for Laplace’s equation. By constructing the approximate value of the unknown boundary function on the side of the rectangle where the nonlocal condition was given, the solution of the multilevel nonlocal problem is defined as a solution of the Dirichlet problem. The uniform estimation of the error of the approximate solution is of order O ( h 2 ), where h is the mesh step. Numerical experiments are presented to support the theoretical analysis made.

In a rectangular domain, we consider the Bitsadze–Samarskii nonlocal boundary value problem for the two-dimensional Poisson equation. The solution of this problem is defined as a solution of the local Dirichlet boundary value problem, by constructing a special method to find a function as the boundary value on the side of the rectangle, where the nonlocal condition was given. Further, the five point approximation of the Laplace operator is used for the realization of the proposed method. Numerical experiments are illustrated in the last section to support the analysis made.

A nonlocal boundary value problem for Laplace’s equation on a rectangle is considered. Dirichlet boundary conditions are set on three sides of the rectangle, while the boundary values on the fourth side are sought using the condition that they are equal to the trace of the solution on the parallel midline of the rectangle. A simple proof of the existence and uniqueness of a solution to this problem is given. Assuming that the boundary values given on three sides have a second derivative satisfying a Hölder condition, a finite difference method is proposed that produces a uniform approximation (on a square mesh) of the solution to the problem with second order accuracy in space. The method can be used to find an approximate solution of a similar nonlocal boundary value problem for Poisson’s equation.

The prominent Russian mathematician Sergei Mikhailovich Nikol’skii, scholar, educator, organizer of science, and academician of the Russian Academy of Sciences, passed away on 9 November 2012, in his 108th year. Sergei Mikhailovich had a long and exemplary life. He was born at Zavod Talitsa in Perm’ Province (now Talitsa in Perm’ Oblast) on 30 April 1905. His father, Mikhail Dmitrievich Nikol’skii, held the rank of forestry specialist 1st class, and worked as a forest warden in Voronezh Province after the October Revolution. He was killed by bandits in 1921. Sergei’s mother, Lyudmila Mikhailovna, was a village teacher. In his teens Sergei began working in forestry, and later at a state farm. After Mikhail’s death the Nikol’skii family moved to Chernigov, where Sergei worked in provincial political education and studied at the Industrial and Agronomical Technical College. In 1925 he was admitted to the Institute of Public Education at Ekaterinoslav (now Dnepropetrovsk), and already during his first year there he became seriously interested in mathematics. In 1929 Nikol’skii was kept on at the Institute as an assistant professor. He regularly attended lectures delivered by A. N. Kolmogorov and P. S. Alexandrov, professors at Moscow State University who periodically visited Dnepropetrovsk. Soon he became one of Kolmogorov’s students, and found in him not merely a scientific advisor but also a general mentor. In 1934–1935 Nikol’skii was sent to Moscow State University to work on his Ph.D. dissertation. He studied, in his own words, from nine o’clock in the morning till nine o’clock in the evening, attending the seminars of Kolmogorov, V. L. Goncharov, A. I. Plessner, L. A. Lyusternik, and others. After defending in 1935 his dissertation on linear equations in Banach spaces, he became head of the Department of the Theory of Functions at Dnepropetrovsk University. The seminar he led there is now regarded as the origin of the Dnepropetrovsk school on the theory of approximation of functions. In January 1941 he became a postdoctoral researcher at the Steklov Mathematical Institute. Before the Second World War he completed work for his D.Sc. degree.

The existence and uniqueness of a classical solution to the nonlocal boundary value problem for Poisson's operator on a two-dimensional rectangular domain is proved in detail by applying the contraction mapping principle.

The Dirichlet problem for Laplace’s equation on an infinite rectangular cylinder is considered. The main goal is to develop a grid method for finding an approximate solution of the Dirichlet problem in a finite part of the infinite cylinder without solving the entire problem. The underlying idea is that the influence of the boundary values on the solution at a fixed point of the domain decreases as the boundary moves away.

In this paper, a homogeneous scheme with 26-point averaging operator for the solution of Dirichlet problem for Laplace’s equation on rectangular parallelepiped is analyzed. It is proved that the order of convergence is O(h 4), where h is the mesh step, when the boundary functions are from C 3, 1, and the compatibility condition, which results from the Laplace equation, for the second order derivatives on the adjacent faces is satisfied on the edges. Futhermore, it is proved that the order of convergence is O(h 6(|lnh| + 1)), when the boundary functions are from C 5, 1, and the compatibility condition for the fourth order derivatives is satisfied. These estimations can be used to justify different versions of domain decomposition methods.

The Dirichlet problem for Laplace’s equation in a rectangular parallelepiped is solved by applying the grid method. A 14-point averaging operator is used to specify the grid equations on the entire grid introduced in the parallelepiped. Given boundary values that are continuous on the parallelepiped edges and have first derivatives satisfying the Lipschitz condition on each parallelepiped face, the resulting discrete solution of the Dirichlet problem converges uniformly and quadratically with respect to the mesh size. Assuming that the boundary values on the faces have fourth derivatives satisfying the Hölder condition and the second derivatives on the edges obey an additional compatibility condition implied by Laplace’s equation, the discrete solution has uniform and quartic convergence with respect to the mesh size. The convergence of the method is also analyzed in certain cases when the boundary values are of intermediate smoothness.

We study the Dirichlet problem for the Laplace equation in an infinite rectangular cylinder. Under the assumption that the boundary values are continuous and bounded, we prove the existence and uniqueness of a solution to the Dirichlet problem in the class of bounded functions that are continuous on the closed infinite cylinder. Under an additional assumption that the boundary values are twice continuously differentiable on the faces of the infinite cylinder and are periodic in the direction of its edges, we establish that a periodic solution of the Dirichlet problem has continuous and bounded pure second-order derivatives on the closed infinite cylinder except its edges. We apply the grid method in order to find an approximate periodic solution of this Dirichlet problem. Under the same conditions providing a low smoothness of the exact solution, the convergence rate of the grid solution of the Dirichlet problem in the uniform metric is shown to be on the order of O(h 2 ln h −1), where h is the step of a cubic grid.

A novel two-stage difference method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. At the first stage, approximate values of the sum of the pure fourth derivatives of the desired solution are sought on a cubic grid. At the second stage, the system of difference equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first stage. The difference equations at the first and second stages are formulated using the simplest six-point averaging operator. Under the assumptions that the given boundary values are six times differentiable at the faces of the parallelepiped, those derivatives satisfy the Hölder condition, and the boundary values are continuous at the edges and their second derivatives satisfy a matching condition implied by the Laplace equation, it is proved that the difference solution to the Dirichlet problem converges uniformly as O ( h 4 ln h −1 ), where h is the mesh size.

A combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped is proposed. At the grid points that are at the distance equal to the grid size from the boundary, the 6-point averaging operator is used. At the other grid points, the 26-point averaging operator is used. It is assumed that the boundary values have the third derivatives satisfying the Lipschitz condition on the faces; on the edges, they are continuous and their second derivatives satisfy the compatibility condition implied by the Laplace equation. The uniform convergence of the grid solution with the fourth order with respect to the grid size is proved

A high accurate difference-analytical method is introduced for the solution of the mixed boundary value problem for Laplace's equation on graduated polygons. The polygon can have broken sections and be multiply connected. The uniform estimate of the error of the approximate solution is of order O(h6), whereas it is of order $O(h^{6}/r_{j}^{p-\lambda _{j}})$ for the errors of p-order derivatives (p=1,2,. . .) in a finite neighborhood of reentry vertices; here, h is the mesh step, rj is the distance from the current point to the vertex in question, $\lambda _{j}=1/(a\alpha _{j})$, and a=1 or 2 depending on the types of boundary conditions. Further, $\alpha _{j}\pi $ is the value of the interior angle at the considered vertex. Numerical experiments are illustrated in section \ref{sec8} to support the analysis made.

Using the method of composite square and polar grids, we construct approximations of the first derivatives of the solution to the Dirichlet problem for the Laplace equation on a polygon and find error estimates for such approximations.

We present and justify difference schemes for obtaining the solution of the Dirichlet problem, its first derivatives and pure second derivatives on a cubic grid with uniform accuracy O (h2), h is a grid step. We propose a method for finding a mixed derivative with accuracy O (h2/ (ρ + h)), where ρ is the distance from a current mesh node to the boundary of a parallelepiped.

We consider the difference scheme of the solution to the Dirichlet problem for the Laplace equation in a cylinder, using the linear Collatz interpolation at near-boundary mesh nodes. The upper uniform bound of the error of the form O(h(2) In h(-1)) obtained by the author in [10] for both the lateral surface of a cylinder and its boundary values of class C-1,C-1 is proved to be unimprovable in order with respect to the grid step h. It is also established that an additional requirement of the existence of the continuous second derivatives in boundary values on the lateral surface of the cylinder does not ensure the convergence of the difference solution with rate h(2) even if the cylinder is circular. We derive the upper uniform bound of the error of order h(2) for the sufficiently smooth lateral surface of the cylinder and the boundary values of class C(2,lambda,)0 < lambda < 1. These boundary values are supposed to satisfy a compatibility condition on the edges of the cylinder, which results from the Laplace equation.