A finite-dimensional problem with several \(M\)-matrices and diagonal maximal monotone operators is studied. The problem includes variational inequalities with \(M\)-matrices. Existence of a unique solution is studied as well as the convergence and geometric rate of convergence for a class of iterative methods. For instance the Schwarz alternating type methods are considered.
Then the author applies the general results to a mesh scheme for a dam problem. Parallel iterative methods are proposed based on domain decomposition. The geometric convergence is proved. All results of this paper are theoretical.