The paper treats the Schwartz alternating method in subspaces for an approximate solution of the Neumann problem for the second-order elliptic equation in a domain which is a union of disjoint subdomains. The original problem is reduced to an equivalent problem at the boundaries of these subdomains; to solve this problem, an iterative method with the spectrally equivalent operator is proposed. The method proves to converge at the rate of geometric progression. This paper is devoted to the application of the Schwartz alternating method in subspaces (4,5,7) to obtaining approximate solutions to Neumann problems for second-order elliptic equations. The original domain where the solution to the boundary value problem is sought is partitioned into disjoint subdomains, and at every step of the iterative process it is necessary to solve Dirichlet problems in each of these disjoint subdomains. This paper differs from (4,5,7) in the fact that the operator of the boundary value problem is not positive definite, and it has a kernel consisting of functions identically constant in the domain. Despite this fact we manage to prove that the Schwartz alternating method in subspaces converges at the rate of geometric progresion. The method is formulated and investigated for functions from the Sobolev space. This method however can be also applied to solving singular systems of mesh equations approximating the boundary value problems (3) mentioned above. In this case, the convergence rate of the method coincides with that of geometric progression whose ratio is independent of the mesh step size. 1. PROBLEM FORMULATION Let Ω be a bounded domain on the plane with the Lipschitz boundary dQ. Let us seek the normal generalized solution to the Neumann boundary value problem 2 θ