Geodesic
The method of collocations and least residuals for three-dimensional Navier-Stokes equations
Numerical methods and programming, Tome 14 (2013) no. 3, pp. 306-322

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The method of collocations and least residuals, which was previously proposed for the numerical solution of two-dimensional Navier-Stokes equations, is extended for the three-dimensional case. In the implemented version of the method, the solution is sought in the form of an expansion in basis solenoidal functions. An overdetermined system of linear algebraic equations is obtained in each cell of the computational grid. This system is solved by the method of rotations. To accelerate the iteration process convergence, a new algorithm is proposed on the basis of Krylov's subspaces. The results of the method's verification confirm its second order of convergence for the velocity vector components. The results of solving the benchmark problem of a lid-driven cubic cavity flow for the Reynolds numbers ${\rm Re} = 100$ and ${\rm Re} = 1000$ are discussed. It is shown that the obtained results are very close to the most accurate results obtained by other authors with the aid of different numerical high-accuracy methods.
Keywords: three-dimensional Navier-Stokes equations; method of collocations and least residuals; cubic cavity flow; overdetermined linear system; Krylov's subspaces. 
@article{VMP_2013_14_3_a4,
     author = {V. P. Shapeev and E. V. Vorozhtsov and V. I. Isaev and S. V. Idimeshev},
     title = {The method of collocations and least residuals for three-dimensional {Navier-Stokes} equations},
     journal = {Numerical methods and programming},
     pages = {306--322},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2013_14_3_a4/}
}
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V. P. Shapeev; E. V. Vorozhtsov; V. I. Isaev; S. V. Idimeshev. The method of collocations and least residuals for three-dimensional Navier-Stokes equations. Numerical methods and programming, Tome 14 (2013) no. 3, pp. 306-322. http://geodesic.mathdoc.fr/item/VMP_2013_14_3_a4/