The paper is devoted to the construction of effective model grid operators (preconditioners) for classes of linear grid systems with positive operators. The suggested constructions lead to estimates of the spectral condition number like \(O(\log^2N)\); the model operators are almost spectrally equivalent to the original ones.
The grid systems under consideration correspond to special nonconforming methods for the Dirichlet problem for Poisson’s equation in the two-dimensional case. The methods are effective means to combine different approximations in the subdomains, but they lead to good approximations only in energy spaces for the subdomains but not in the whole original domain.