A Block Monotone Iterative Method for Numerical Solutions of Nonlinear Elliptic Boundary Value Problems

作者：Wang, Yuan-Ming^*; Liang, Cui-Xia; Agarwal, Ravi P.

来源：Numerical Methods for Partial Differential Equations, 2011, 27(3): 680-701.

DOI：10.1002/num.20546

• finite difference system
• elliptic boundary value problem
• block monotone iterative method
• parallel computation
• geometric convergence
• upper and lower solutions

The aim of this article is to develop a new block monotone iterative method for the numerical solutions of a nonlinear elliptic boundary value problem. The boundary value problem is discretized into a system of nonlinear algebraic equations, and a 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 can be computed in a parallel fashion and converge monotonically to a maximal solution or a minimal solution of the system. Three theoretical comparison results are given for the sequences from the proposed method and the block Jacobi monotone iterative method. The comparison results show that the sequence from the proposed method converges faster than the corresponding sequence given by the block Jacobi monotone iterative method. A simple and easily verified condition is obtained to guarantee a geometric convergence of the block monotone iterations. The numerical results demonstrate advantages of this new approach.

• 出版日期2011-5
• 单位华东师范大学; 上海师范大学; 中国石油大学（北京）; 上海大学