On the solution of the polynomial systems arising in the discretization of certain ODEs
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 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.
stationary solution neumann boundary condition homotopy continuation two-point boundary-value problem finite differences polynomial system solving condition number complexity