A simple and efficient error analysis for multi-step solution of the Navier-Stokes equations. (Q2779088)

The paper is concerned with numerical solution of incompressible Navier-Stokes problem in two spatial dimensions using a projection method. The aim is to compute steady-state solutions as limites of non-stationary solutions. For the time discretisation, a backward Euler scheme with small time steps is employed.NEWLINENEWLINEUnfortunately, the paper's title is quite misleading, at least for a numerical analyst: Neither an ``error analysis'' is presented nor a ``multi-step'' method is considered. What is presented is an error indicator that relies upon measuring of how well the steady-state momentum equation is fulfilled.NEWLINENEWLINEThe author compares two approaches by \textit{G. Comini} and \textit{S. Del Giudice} [Numer. Heat Transfer 5, 463ff (1982)] and \textit{J. K. Dukowicz} and \textit{A. S. Dvinsky} [J. Comp. Phys. 102, No. 2, 336--347 (1992; Zbl 0760.76059)]. The difference between both remains unclear to the reader, as the author states on p. 592 ``Following the work of Dukowicz and Dvinsky \dots{} we develop \dots'' and later continues on p. 593 ``This formulation is exactly that presented by Comini and Del Guidice''. However, the method under consideration uses linear finite elements for both the velocity and pressure. The system of discrete equations is then solved by approximate factorisation.NEWLINENEWLINETwo numerical examples are discussed: the driven lid cavity problem for Reynolds numbers between 100 and 3200, and the flow over a backstep for Reynolds number 150.NEWLINENEWLINEAs the usage of mathematical language is sometimes not quite clear (e. g. on p. 591 the author gives a rather misleading interpretation of Babuška-Brezzi condition when saying ``the pressure must be of one polynomial order lower than that of the velocity''), reading the paper is somewhat difficult.