On the stability of differencing schemes with smoothing operators (Q1176911)

The author considers two-level difference schemes approximating the Cauchy problem for a linear differential equation with variable operator in a Banach space. These schemes are constructed according to a rather general discretization method. In his previous work the author obtained a priori estimates for these schemes in the case of a constant operator. Here the results are translated to the case of a variable operator. A stability theorem for the proposed class of difference schemes is proved. General results are applied to the solutions of parabolic partial differential equations with variable coefficients.