Abstract In this paper we investigate the stability of two-level operator-difference
schemes in Hilbert spaces under perturbations of operators, the initial condition and
right hand side of the equation. A priori estimates of the error are obtained in time-
integral norms under some natural assumptions on the perturbations of the operators.
Abstract In this work, a stability of three-level operator-difference schemes on
nonuniform in time grids in Hilbert spaces is studied. A priori estimates of a long
time stability (for t → ∞) in the sense of the initial data and the right-hand side
are obtained in different energy norms without demanding the quasiuniformity of the
grid. New difference schemes of the second order of local approximation on nonuniform
grids both in time and space on standard stencils for parabolic and wave equations are
adduced.
Nowadays the general theory of operator-difference schemes with operators acting in Hilbert spaces has been created for investigating the stability of the difference schemes that approximate linear problems of mathematical physics. In most cases a priori estimates which are uniform with respect to the t norms are usually considered. In the investigation of accuracy for evolutionary problems, special attention should be given to estimation of the difference solution in grid analogs of integral with respect to the time norms. In this paper a priori estimates in such norms have been obtained for two-level operator-difference schemes. Use of that estimates is illustrated by convergence investigation for schemes with weights for parabolic equation with the solution belonging to [Formula: see text].
Nonclassical problem for ultraparabolic equation with nonlocal initial condition with respect to one time variable is studied in abstract Hilbert spaces. We define the space of square integrable vector-functions with values in Hilbert spaces corresponding to the variational formulation of the nonlocal problem for ultraparabolic equation and prove trace theorem, which allows one to interpret initial conditions of the nonlocal problem. We obtain suitable a priori estimates and prove the existence and uniqueness of solution of the nonclassical problem and continuous dependence upon the data of the solution to the nonlocal problem. We consider an application of the obtained abstract results to nonlocal problem for ultraparabolic partial differential equation with second-order elliptic operator and obtain well-posedness result in Sobolev spaces.
A priori estimates of smoothness of solutions to difference Bellman equations with linear and quasi-linear operators