Inexact monotone methods for solving nonlinear elliptic problems

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-