Navier--Stokes equations in rotation form: A robust multigrid solver for the velocity problem (Q2780598)

The paper is devoted to numerical methods for Navier-Stokes equations (incompressible flows) with special representation of the nonlinear term. The importance of such choice is known for a long time and usually Temam's form is used. The authors prefer to write the nonlinear term in the form \((\operatorname {curl}\vec u)\times\vec u\) -- it has the same important properties of Temam's form (this is not mentioned by the authors). Then the linearized equations (written in the block form) contain the strongly elliptic operator defined by the system \(-\nu\Delta\vec u+\vec w\times\vec u+c\vec u=\vec f \;(c\geq 0)\) with the standard boundary conditions \(\vec u|_{\Gamma}=0\) on the boundary of a convex polygonal domain. The main results of the paper are connected with multigrid methods for the discretized problem (standard triangular spline approximations are used), but the authors also present certain facts about the differential problem and convergence of the numerical method. It should be noted that the theory of multigrid methods under consideration is based on the assumption that \(\vec u\in [H^2(\Omega)]^2\) and that special attention is paid to the dependence of convergence on the parameters \(h,\nu,c\). The authors claim that the convergence is uniform (robust). Results of numerical experiments are described. NEWLINENEWLINENEWLINEIt should be noted that asymptotically (with respect to \(h\)) optimal algorithms for similar elliptic two and three-dimensional problems were constructed under more general conditions and can be found in the reviewer's book [Optimization in solving elliptic problems (1996; Zbl 0852.65087)]; they were based on the theory of algebraic multigrid methods. Also it is of importance that for the discretized Navier-Stokes equations (in Temam's form) effective iterative methods, not dealing with the Schur-Crouzey operator, are known (see the cited reviewer's book; the authors mention only ``Uzawa iterations'').