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Dirichlet's problem is considered for a weakly nonlinear elliptic equation of second order. The implementation of the standard finite element method with piecewise linear elements on triangles results in the system of nonlinear algebraic equations. To solve it a cascade organization of the conjugategradient method is used on a sequence of nested grids and it gives a simple version of the multigrid without preconditioning and projection onto the coarser grid. The cascade algorithm begins at a comparatively coarse grid, where the number of unknowns in a discrete nonlinear system is several orders less than this number at the finest grid. Therefore we suppose that this system is previously solved with sufficiently high accuracy and appropriate computational complexity. At each finer grid the nonlinear system is linearized by Newton's method with frozen derivative' and is approximately solved by the conjugate-gradient method; the initial guess is obtained by interpolation of an approximate solution from the previous coarser grid. It is proved that this cascade algorithm has the same optimal order of computational complexity as the traditional multigrid. The cascadic conjugate-gradient method was originally presented by P. Deuflhard in [4] and [5], where the high rate of convergence was shown computationally. The optimal computational complexity of this algorithm for the second-order elliptic equation with smooth solution is proved in [8,9]. The optimal complexity for the elliptic equation is established in [1,10], its solution is not sufficiently smooth since the domain has interior angles in excess of π. This proof is generalized [11] to the problem in the domain with curvilinear boundary, where the nested Galerkin subspaces are used without strict nestedness of triangulation sequence. Besides, F. A. Bornemann [l] obtained estimates of the convergence rate of other iterative smoothing operators in twoand three-dimensional boundary value problems for the second-order elliptic equation. In the two-dimensional case of the smoothing operators, which he analysed, only the conjugate-gradient method gave the optimal arithmetic complexity. The Jacobi-type method with special iterative parameters also gives this optimality (see [8,9]). In the three-dimensional case several more smoothing operators ensure the optimal complexity. In the above-mentioned papers the linear boundary value problems were considered. In the present paper we study the application of the cascadic conjugate-gradient method and the Jacobi-type one with special parameters to a sequence of grid problems obtained by linearization of a weakly nonlinear elliptic equation by Newton's method with 'frozen derivative'. It was proved that the presented algorithm has the same optimal order of computational complexity as the traditional multigrid one [6,12]. * Institute of Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk 660036, Russia The first author's research was partially supported by grant No. 1/72342 of Volkswagen Foundation and Grant No. 98-01-00704 of the Russian Foundation for the Basic Research. The other author's research was performed during stay at the Institute of Numerical Mathematics of Dresden Technical University. 60 L. V. Gifyova and V. V. Shaidurov 1. FORMULATION OF THE DIFFERENTIAL PROBLEM AND THE FINITE ELEMENT METHOD SCHEME On the bounded convex polygon Ω c R with the boundary Γ we consider the problem Au = f(x,u) in Ω (1.1) u = 0 on Γ (1.2) where the function /(x,w) belongs to <7(Ω χ R) and satisfies the constraints:

A cascade algorithm for solving a discrete analogue of weakly nonlinear elliptic equation