EFFICIENT SOLUTION OF CERTAIN SPARSE POLYNOMIAL SYSTEMS DERIVED FROM DIFFERENTIAL EQUATIONS

DRATMAN, EZEQUIEL

Paraty. Río de Janeiro. Barsil.

Workshop; Complexity of Algorithms for Solving Equations; 2012

UFRJ

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 or 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 ε-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. In the remaining cases we prove that there exist spurious solutions. From these results we obtain a complete outlook of the comparison between the stationary solutions of the differential problem under consideration and its discretization.