Numerical solution of nonlinear elliptic systems by block monotone iterations

We present numerical methods for solving a coupled system of nonlinear elliptic problems, where reaction functions are quasimonotone nondecreasing. We utilize block monotone iterative methods based on the Jacobi and Gauss--Seidel methods incorporated with the upper and lower solutions method. A convergence analysis and the theorem on uniqueness of solutions are discussed. Numerical experiments are presented. References Boglaev, I., Monotone iterates for solving systems of semilinear elliptic equations and applications, ANZIAM J, Proceedings of the 8th Biennial Engineering Mathematics and Applications Conference, EMAC-2007, 49(2008), C591C608. doi:10.21914/anziamj.v49i0.311 Pao, C. V., Nonlinear parabolic and elliptic equations, Springer-Verlag (1992). doi:10.1007/978-1-4615-3034-3 Pao, C. V., Block monotone iterative methods for numerical solutions of nonlinear elliptic equations, Numer. Math., 72(1995), 239262. doi:10.1007/s002110050168 Samarskii, A., The theory of difference schemes, CRC Press (2001). https://www.crcpress.com/The-Theory-of-Difference-Schemes/Samarskii/p/book/9780824704681 Varga, R. S., Matrix iterative analysis, Springer-Verlag (2000). doi:10.1007/978-3-642-05156-2