Newton’s method and a mesh-independence principle for certain semilinear boundary-value problems

We exhibit an algorithm which computes an ϵϵ-approximation of the positive solutions of a family of boundary-value problems with Neumann boundary conditions. Such solutions arise as the stationary solutions of a family of semilinear parabolic equations with Neumann boundary conditions. The algorithm is based on a finite-dimensional Newton iteration associated with a suitable discretized version of the problem under consideration. To determine the behavior of such a discrete iteration we establish an explicit mesh-independence principle. We apply a homotopy-continuation algorithm to compute a starting point of the discrete Newton iteration, and the discrete Newton iteration until an ϵϵ-approximation of the stationary solution is obtained. The algorithm performs roughly O((1/ϵ)1/2)O((1/ϵ)1/2) flops and function evaluations.