Numerical approximation of the non-negative solution of a type of elliptic or parabolic semi-linear equations

The aim of this paper is to present a numerical method for calculating an approximation to the non-negative solution a type of semi-linear elliptic and parabolic equations. First is introduce d a theoretical result about existence of a positive solution in the elliptic case. Then a numerical algorithm using Newton’s method and domain decomposition method is developed following the theoretical analysis.

Then we applied the same technic to compute a positive solution of semi linear parabolic equations

In the case of the blows-up of the solution, we will give an estimate of the maximum time of existence of the numerical solution. This estimate is a precise approximation of the maximal time existence of the analytical solution. We have also adapted the waveform algorithm to the problem considered. In all cases, the simulations converge, illustrating the performance of the algorithms studied and the consistency of the results with the theory.