Newton's method and a mesh independence principle for certain semilinear boundary value problems.

EZEQUIEL DRATMAN; GUILLERMO MATERA

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2016 vol. 292 p. 188 - 212

0377-0427

We exhibit an algorithm which computes an $epsilon$--approximation of 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 $epsilon$--approximation of the stationary solution is obtained. The algorithm performs roughly $mathcal{O}((1/epsilon)^{1/2})$ flops and function evaluations.