Inexact block monotone methods for solving nonlinear elliptic problems

The paper deals with numerical solving of semilinear elliptic problems based on the method of upper and lower solutions. Inexact block monotone iterative methods are constructed, where monotone linear systems are solved by the block Jacobi or block Gauss–Seidel methods only approximately. The inexact block monotone methods combine the quadratic monotone iterative method at outer iterations and the block iterative methods at inner iterations, and possess global monotone convergence. Results of numerical experiments, implemented in the framework of an inexact Newton method, are presented, where iteration counts and CPU times of the inexact block monotone methods are compared with the block monotone iterative methods, whose convergence rate is linear.