On energy estimates for preconditioning grid parabolic problems

The paper deals with studying of some preconditioning operators providing unconditional convergence of difference schemes for solving parabolic problems. Three examples of such operators are considered. There are a preconditioner of the domain decomposition method of the Neumann-Dirichlet type and two preconditioners of the fictitious domain methods for the Neumann and the Dirichlet boundary value problems respectively.