Asynchronous multisplitting two-stage iterations for systems of weakly nonlinear equations

For the large sparse systems of weakly nonlinear equations arising in the discretizations of many classical differential and integral equations, this paper presents a class of asynchronous parallel multisplitting two-stage iteration methods for getting their solutions by the high-speed multiprocessor systems. Under suitable assumptions, we study the global convergence properties of these asynchronous multisplitting two-stage iteration methods. Moreover, for this class of new methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some reasonable assumptions when the involved nonlinear mapping is only assumed to be directionally differentiable. Numerical computations show that our new methods are feasible and efficient for parallely solving the system of weakly nonlinear equations.