We numerically solving semilinear elliptic problems with the method of upper and lower solutions. Inexact monotone iterative methods are constructed, where monotone linear systems are solved by the Jacobi or Gauss--Seidel methods only approximately. The inexact monotone methods combine the quadratic monotone iterative method at outer iterations and the Jacobi or Gauss--Seidel methods at inner iterations, and possess global monotone convergence. Results of numerical experiments are presented. References B. Abraham and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences . Academic Press, New York, 1979. doi:10.1137/1.9781611971262 I. Boglaev, Uniform quadratic convergence of monotone iterates for semilinear singularly perturbed elliptic problems. Lecture Notes in Computational Science and Engineering 81:37–46, 2011. doi:10.1007/978-3-642-19665-2_5 I. Boglaev, Monotone relaxation iterates and applications to semilinear singularly perturbed problems. Int. J. Numer. Anal. Mod.(B) 2:402–414, 2011. http://www.math.ualberta.ca/ijnamb/Volume-2-2011/No-4-11/2011-04-08.pdf I. Boglaev, An inexact monotone method for solving semilinear parabolic problems. Appl. Math. Comput. 219:3253–3263, 2012. doi:10.1016/j.amc.2012.09.067 R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods. SIAM J. Numer. Anal. 19:400–408, 1982 doi:10.1137/0719025 S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method. SIAM J. Sci. Comput. 17:16–32, 1996. doi:10.1137/0917003 P. Knabner and L. Angerman, Numerical Methods for Elliptic and Parabolic Partial Differential Equations . Springer, New York, 2003. doi:10.1007/b97419 C. V. Pao, Nonlinear Parabolic and Elliptic Equations . Springer, New York, 1992. doi:10.1007/978-1-4615-3034-3 C. V. Pao, Accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems. Computers Math. Applic. , 46:1535–1544, 2003. doi:10.1016/S0898-1221(03)00381-X