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 epsilon-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) 2013 Elsevier Inc. All rights reserved.
stationary solution homotopy continuation two-point boundary-value problem finite differences condition number complexity