Fourth-Order Compact Finite Difference Methods and Monotone Iterative Algorithms for Quasi-Linear Elliptic Boundary Value Problems

This paper is concerned with numerical methods for a class of two-dimensional quasi-linear elliptic boundary value problems. A compact finite difference method with a nonisotropic mesh is proposed for the problems. The existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions, and two sufficient conditions for the uniqueness of the solution are also given. The optimal error estimate in the discrete $L^{\infty}$ norm is obtained under certain conditions. The error estimate shows the fourth-order accuracy of the proposed method when two spatial mesh sizes are proportional. By using an upper solution or a lower solution as the initial iteration, an “almost optimal” Picard type of monotone iterative algorithm is presented for solving the resulting nonlinear discrete system efficiently. Applications using two model problems give numerical results that confirm our theoretical analysis.