Monotone iterates for solving coupled systems of nonlinear parabolic equations

Numerical solutions of coupled systems of nonlinear elliptic equations

“…(n) }, generated by (9), (10) with, respectively,Ū (0) =Ũ and U (0) =Û, are ordered upper and lower solutions to (2) and converge monotonically to U andŪ, such that…”

“…We say that (Ū, U) is a quasi-solution of the difference scheme (2), if it satisfies the boundary conditions in (2) and the coupled equations…”

“…Thus, this approach may be considered as a variant of a block Gauss-Seidel iteration (or in the parallel context as a multicolored algorithm) for the subdomains with a DirichletDirichlet coupling through the interface variables. In this article, we extend the monotone domain decomposition algorithms from [2,3] to the system of nonlinear elliptic equations (1) with mixed quasi-monotone reaction functions.…”

“…In [2] and [3], for solving systems of two elliptic equations with, respectively, quasi-monotone nondecreasing and nonincreasing reaction functions, iterative algorithms which combine the monotone approach and an iterative domain decomposition method based on the Schwarz alternating procedure, have been proposed. The computational domain is partitioned into many nonoverlapping subdomains with interface γ .…”

“…Unlike scalar elliptic equations, the application of this method to coupled systems is more complicated especially when the reaction functions are mixed quasi-monotone. This method is used in the case of two elliptic equations in [1] for reaction functions of quasimonotone nonincreasing type and in [2] for reaction functions of quasi-monotone nondecreasing type. In this study, we extend the monotone iterative method to the system of nonlinear elliptic equations (1) with mixed quasi-monotone reaction functions.…”