On the solution of the polynomial systems arising in the discretization of certain ODEs

EZEQUIEL DRATMAN; GUILLERMO MATERA

COMPUTING

SPRINGER WIEN

Lugar: Viena; Año: 2009 vol. 85 p. 301 - 337

0010-485X

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 ``continuous´´ equation. 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.