Sufficient conditions for the coefficient stability of operator-difference schemes. (Q1432322)

Introducction: In some differential problems, the coefficients of equations are specified approximately (e.g., they may be obtained with the use of a computational algorithm, physical measurements, etc.). For this reason, the problem of studying schemes with perturbed coefficients is very important. We say that a difference scheme is strongly stable if it is stable with respect to the initial data, right-hand side, and coefficients. At present, studies of strong stability for evolution differential-operator equations and operator-difference schemes are being started. In the monographs of \textit{A. A. Samarskij} and \textit{A. V. Gulin} [Stability of difference schemes. (1973; Zbl 0304.65037)] and of \textit{A. A. Samarskij} [Theory of difference schemes (1977; Zbl 0462.65055)] it is shown that stability with respect to the initial data implies stability with respect to the right-hand side if the norms are consistent in a certain sense. In this paper, we prove that stability with respect to the initial data is sufficient for the stability of the solution to an operator-difference scheme with respect to the operator coefficients.