On the convergence of classical and multilevel variants of the Uzawa and Arrow–Hurwicz algorithms related to the LBB condition

In this paper we propose a systematic study of classical and multilevel variants of the Uzawa and Arrow–Hurwicz methods. The multilevel methods are obtained from the classical ones by the introduction of multilevel inner iterations to calculate the solution of the first equation instead of its exact calculation as in the classical Uzawa or Arrow–Hurwicz methods. In our study, an essential role is played by the LBB condition. For classical and multilevel variants of the Uzawa and Arrow–Hurwicz algorithms, we prove theorems which give the convergence conditions of the methods and explicit formulas of the convergence rates. On the basis of these results we compare the convergence conditions and the convergence rates of the classical methods with those of their corresponding ones in the multilevel methods. Concerning the Uzawa methods, we prove that, the limit of the convergence condition and the convergence rate of the multilevel method, when the number of the inner iterations tends to infinity, coincide with those of the classical one. Also, from the dependence of the convergence rate on the number of inner iterations of the multilevel method, we conclude that, the multilevel method with a small number of inner iterations converges better than the classical one. For the Arrow–Hurwicz methods we found that for a large number of inner iterations of the multilevel algorithm, the convergence condition of the multilevel method coincides with that of the classical method and the convergence rate of the multilevel method is equal to or smaller than that of the classical method. Finally, the behavior of the introduced methods is investigated by numerical experiments carried out for the driven-cavity Stokes problem and they confirm the theoretical results.