Efficient approximation of the solution of certain nonlinear reaction-diffusion equations with large 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 large 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 E-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. (C) 2012 Elsevier B.V. All rights reserved.