Efficient approximation of the solution of certain nonlinear reaction–diffusion equations with small absorption

We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if the absorption is small enough  , compared with the flux in the boundary, there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the “continuous” equation. Furthermore, we exhibit an algorithm computing an εε-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.