Root mean square error estimates for the projection-difference method for the approximate solution of a parabolic equation with a periodic condition for the solution

We construct an approximate solution of an abstract linear parabolic equation in a separable Hilbert space with a periodic condition for a solution by the projection-difference method. We use the Galerkin method for the spatial variables and the implicit Euler discretization for time. We obtain root mean square estimates of the error of approximate solutions that are effective both in time and spatial variables; these estimates imply the convergence of approximate solutions to an exact solution and allow one to find the convergence rate.