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 diffusion is large enough, then there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the boundary-value problem under consideration. Furthermore, in this case we exhibit an algorithm computing an [straight epsilon]-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is polynomial in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required. [PUBLICATION ABSTRACT]
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