Numerical solution of nonlinear elliptic systems by block monotone iterations

Abstract

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), C591–C608. 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), 239–262. 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