Convergence of the projection-difference method for the approximate solution of a smoothly solvable parabolic equation with a weighted integral condition

We search for an approximate solution of an abstract linear parabolic equation in a Hilbert space with a nonlocal weighted integral condition by the projection-difference method and the implicit Euler method in time. The approximation of the problem with respect to spatial variables is oriented to the finite element method in the case of arbitrary projection subspaces under an additional smoothness condition. Estimates of errors of approximate solutions are established, the convergence of approximate solutions to the exact solution is proved, and the convergence rate is estimated.