The paper is devoted to iterative methods of Uzawa type for grid analogues of the Dirichlet boundary value problem for the stationary Stokes system. The authors believe that addition of the operator \(-\rho \nabla\)div to the original operator \(-\nu \Delta\) impoves the situation when the constant \(\beta>0\) in the well-known inf-sup condition is small. Their iterations include the exact soluions of grid systems with the “new” operator (actually these systems are very hard to solve and were investigated in many papers dealing with elasticity problems). In the center of the analysis the authors put the problem of the choice of the relaxation parameter \(\alpha>0\) in the grid analog of the iteration \(p^{n+1}=p^n-\alpha\nu \text{ div } u^{n+1}\). They write that the optimal choice is \(\alpha=\sigma\equiv 1+\nu^{-1}\rho\in(0,2\sigma)\) (regardless of \(\beta\)) and that it yields the contraction number \(q\equiv[1-\beta^2]^{1/2}\) (see (1.8) and remark 1.1). But this directly contradicts the statement in the summary that they obtained the convergence with \(q=\kappa\beta\).
It seems that the authors did not know about many results dealing with effective iterative methods for the Stokes problem [part of the references can be found in the book of E. G. D’yakonov, Optimization in solving elliptic problems (1996; Zbl 0852.65087)]. But what is really surprising is the fact that the inequality \(\beta\leq 1\) is considered as a new one and that it can be of any help in the investigation. Several inaccuracies are obvious: for example, the definition of \(P_h\) (p. 696) does not include orthogonalization to 1; an open polygon is considered as a union of several closed elements.