A nonlinearity lagging for the solution of nonlinear steady state reaction diffusion problems

This paper concerns with the analysis of the iterative procedure for the solution of a nonlinear reaction diffusion equation at the steady state in a two dimensional bounded domain supplemented by suitable boundary conditions. This procedure, called Lagged Diffusivity Functional Iteration (LDFI)-procedure, computes the solution by "lagging'' the diffusion term. A model problem is considered and a finite difference discretization for that model problem is described.Furthermore, properties of the finite difference operator are proved. Then, sufficient conditions for the convergence of the LDFI-procedure are given. At each stage of the LDFI-procedure a weakly nonlinearalgebraic system has to be solved and the simplified Newton-Arithmetic Mean method is used. This method is particularly well suited for implementation on parallel computers.Numerical studies show the efficiency, for different test functions, of the LDFI-procedure combined with the simplified Newton-Arithmetic Mean method. Better results are obtained when in the reaction diffusion equation also a convection term is present.