**5. Conclusions**

In our previous theoretical and numerical studies of the MGM with the STS-based smoothers, the stationary (and nonstationary) linear diffusion–convection equation with dominant convection was considered as a test problem [13,14]. All theoretical results and calculations were performed using geometric MGM. Here, we first use the HSS-method as the smoother in the algebraic MGM for solving the unsteady Navier–Stokes equations. It is supposed to further prove the theoretically smoothing properties of the HSS iteration methods and to prove the convergence of the MGM with the corresponding smoothers. In addition, theoretical and numerical results should be obtained for the MGM with the STS-based smoothers for the Navier–Stokes problem. The PMIS algorithm was not chosen by us by chance. Preliminary testing of it on this model problem showed its robustness. In addition, the PMIS algorithm allows for natural parallelization, unlike the RS algorithm. These properties of the PMIS algorithm seem promising to us for the further study of the three-dimensional Navier–Stokes equations using parallel computing.

**Author Contributions:** Conceptualization, G.M.; methodology, G.M., T.M.; validation, E.A.; writing—original draft preparation, T.M., Z.-Q.W.; writing—review and editing, T.M.; software, V.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by RFBR, grant N19-51-53013 GFENa, and Ministry of Science and Higher Education of the Russian Federation (basic part, project N1.5169.2017/8.9).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Abbreviations**

The following abbreviations are used in this manuscript:

