The author considers the family of two-layer schemes of the kind \[ B_\tau \frac{\overline{u}^{n+1}- \overline{u}^n}{\tau}+ A\overline{u}^n= \overline{f}^n, \qquad n=0,1,\dots,N-1, \tag{1} \] where \(A\) is a positive semidefinite matrix, \(B_\tau= I+\sigma\tau B\) is a positive definite matrix, \(I\) is the identity matrix, \(\sigma>0\). The matrices \(A\) and \(B\) (the preconditioning operator) are symmetric. Instead of the ordinary assumptions about energy equivalence of the operators \(A\) and \(B\) the author uses much weaker conditions: \[ -\alpha\langle A\overline{u}, \overline{u}\rangle\leq \langle B\overline u,\overline u\rangle\leq \beta\langle A\overline{u}, \overline{u}\rangle, \qquad \gamma\langle A_\tau\overline{u}, \overline{u}\rangle\leq\langle B_\tau\overline{u}, \overline{u}\rangle, \] where \(A_\tau= I+\sigma\tau A\), \(\alpha, \beta, \gamma\) are positive numbers. Examples with such a matrix, which is connected with the domain decomposition method and with the fictitious domain method, are presented.