Monotone iterations for numerical solutions of nonlinear elliptic partial differential equations

摘要

Employing a weighted difference scheme and the method of upper-lower solutions, we develop a monotone iterative scheme for the computation of solutions of nonlinear elliptic partial differential equations. In each iteration, we need only solve a linear system of which the coefficient matrix is an M-matrix. Generally, this matrix is nonsymmetric in contrast to the midly nonlinear problems. However, the squared conjugate gradient (CGS) method together with the diagonal preconditioning technique are proven to be very effective. Numerical results that are given for the nonlinear Burger's equation clearly demonstrate the advantages of our method.