Block monotone iterative methods for solving coupled systems of nonlinear elliptic problems

TL;DR

This paper develops and analyzes block monotone iterative methods, based on Jacobi and Gauss-Seidel schemes, for efficiently solving coupled nonlinear elliptic problems, ensuring convergence and solution existence.

Contribution

It introduces a novel approach combining block monotone iterations with upper-lower solution methods for coupled nonlinear elliptic systems.

Findings

Block Gauss-Seidel converges at least as fast as block Jacobi.

Monotone sequences guarantee existence of solutions.

Methods are applicable to quasimonotone nondecreasing reaction functions.

Abstract

This paper investigates numerical methods for solving coupled system of nonlinear elliptic problems. We utilize block monotone iterative methods based on Jacobi and Gauss--Seidel methods to solve difference schemes which approximate the coupled system of nonlinear elliptic problems. In the view of upper and lower solutions method, two monotone upper and lower sequences of solutions are constructed, where the monotone property ensures the theorem on existence of solutions to problems with quasimonotone nondecreasing reaction functions. Construction of initial upper and lower solutions is presented. The sequences of solutions generated by the block Gauss--Seidel method converge not slower than by the block Jacobi method.Mohamed