The strong-norm convergence of a projection-difference method of solution of a parabolic equation with the periodic condition on the solution

A smooth soluble abstract linear parabolic equation with the periodic condition on the solution is treated in a separable Hilbert space. This problem is solved approximately by a projection-difference method using the Galerkin method in space and the implicit Euler scheme in time. Effective both in time and in space strong-norm error estimates for approximate solutions, which imply convergence of approximate solutions to the exact solution and order of convergence rate depending of the smoothness of the exact solution, are obtained.