Parallel Block Monotone Iterative Method For Numerical Solutions Of Nonlinear Elliptic Boundary Value Problems

A parallel block monotone iterative method for the numerical solutions of a nonlinear elliptic boundary value problem is presented. The boundary value problem under consideration is discretized into a system of nonlinear algebraic equations, and the parallel block monotone iterative method is established for the system using an upper solution or a lower solution as the initial iteration. The sequence of iterations is shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a parallel computational algorithm as well as a sufficient condition for the uniqueness of the solution. Various theoretical comparison results for the sequences from the proposed method and the block Jacobi iterative method are given. The comparison results show that the convergence of the sequence from the proposed method is faster. A simple and easily verified condition is obtained to guarantee a geometric convergence of the parallel block monotone iterations. The numerical results demonstrate the high efficiency and advantages of this new approach.