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This is a review of the main results in computational mathematics that were obtained by the eminent Russian mathematician Alexander Andreevich Samarskii (February 19, 1919 – February 11, 2008). His outstanding research output addresses all the main questions that arise in the construction and justification of algorithms for the numerical solution of problems from mathematical physics. The remarkable works of A.A. Samarskii include statements of the main principles re- quired in the construction of difference schemes, rigorous mathematical proofs of the stability and convergence of these schemes, and also investigations of their algorith- mic implementation. A.A. Samarskii and his collaborators constructed and applied in practical calculations a large number of algorithms for solving various problems from mathematical physics, including thermal physics, gas dynamics, magnetic gas dynam- ics, plasma physics, ecology and other important models from the natural sciences.

The time discretisation of the initial-value problem for a first-order evolution equation by the two-step backward differentiation formula (BDF) on a uniform grid is analysed. The evolution equation is governed by a time-dependent monotone operator that might be perturbed by a time-dependent strongly continuous operator. Well-posedness of the numerical scheme, a priori estimates, convergence of a piecewise polynomial prolongation, stability as well as smooth-data error estimates are provided relying essentially on an algebraic relation that implies the G-stability of the two-step BDF with constant time steps.

We propose a new analytical-numerical method with an embedded convergence control mechanism for solving nonlinear operator differential equations. The method provides the exponential convergence rate. A numerical example confirms the theoretical results.

The paper deals with difference schemes for the heat-conduction equa- tion with nonlocal boundary conditions containing two real parameters. Such schemes have been investigated for some special parameter values, but the general case was not considered previously. The eigenvalue problem arises as a result of variable division and is solved here explicitly. The so-called reality domains were selected on the plane for which all eigenvalues and eigenfunctions are real. It was demon- strated that the difference schemes in question are symmetrizable in reality domains, that is their transition operators are similar to self-adjoint ones. The necessary and sufficient stability conditions for difference schemes under consideration are obtained with respect to the initial data in the specially constructed norm. The equivalence of the above-mentioned norm to the grid L2-norm has been proved.

An initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.