A note on the iterative solution of nonlinear steady state reaction diffusion problems

This report concerns with the numerical solution of nonlinear reaction diffusion equations at the steady state in a two dimensional bounded domain supplemented by suitable boundary conditions. When we use finite differences or finite element discretizations, the nonlinear diffusion equation subject to Dirichlet boundary conditions can be transcribed into a nonlinear system of algebraic equations. In the case of finite differences, the matrix that arises from the discretization of the diffusion (and/or convection) term satisfies properties of monotonicity.This report is divided into two parts (chapters): the first part deals with the solution of a weakly nonlinear reaction diffusion equation while in the second part, the solution of a strongly nonlinear reaction diffusion equation is computed by an iterative procedure that “lags” the diffusion term. This procedure is called Lagged Diffusivity Functional Iteration (LDFI)–procedure.In the first part the weakly nonlinear algebraic system arising from the discretization is solved by a simplified Newton method comkbined with the Arithmetic Mean method that is an iterative method, suited for parallel computers, for the solution of large sparse linear systems. This inner-outer iteration process gives a two-stage iterative method.Results concerning the global and monotone convergence for the two-stage iterative method have been reported. Furthermore, numerical experiments show the efficiency of the two-stage iterative method, especially for a dominant convection term, confirming the well known results for the linear case.In the second part the LDFI-procedure for the solution of the strongly nonlinear reaction diffusion equation is analyzed. A model problem is considered and a finite difference discretization for that model problem is described. Furthermore, in the report, 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 nonlinear algebraic system has to be solved and the simplified Newton-Arithmetic Mean method is used. Numerical studies show the efficiency for different test functions of the LDFI-procedure combined with the simplified Newton-Arithmetic Mean method. Better result are obtained for dominant convection coefficients according with the linear and the weakly nonlinear cases.