Estimates of the rate of convergence of projective and projective-difference methods for weakly solvable parabolic equations

We consider a weakly solvable parabolic problem in a separable Hilbert space. We seek approximations to the exact solution by projective and projective-difference methods. In this connection the discretization of the problem with respect to the spatial variables is carried out by the semidiscrete method of Galerkin, and with respect to time by the implicit method of Euler. In this paper we establish a coercive mean-square error estimate for the approximate solutions. We illustrate the effectiveness of these estimates with parabolic equations of second order with Dirichlet or Neumann boundary conditions in projective subspaces of finite element type.