ABSTRACT
Solving large elliptic grid systems resulting from discretization of elliptic boundary value problems on the basis, for example, of finite-element or difference methods, is one of the most important problems of computational mathematics. Iterative methods are an indispensable tool in dealing with this problem, and a very large number of investigations have been devoted to the questions of improving their theoretical and practical efficiency for finding appropriate approximations to the desired solutions. We only mention publications connected (to some degree) with construction of asymptotically optimal or nearly optimal preconditioners and some of their generalizations for nonsymmetric or nonlinear operators (see references 1-45). The usefulness of such iterative methods for effective implementation on modern vector and parallel computers is widely recognized. In the case of a complicated nonlinear operator, L, even the problem of evaluating a residual, rn = L(u n ) − f, with a given iterate, u n , and the right-hand term, f, needs considerable computational work. As usual, it will be characterized by the number of required arithmetical operations. Therefore, for such problems it is especially reasonable not only to apply an iterative method with the rate of convergence independent of the grid (with a parameter h) but also to try to decrease the number of iterations required to obtain the desired accuracy.
