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

作者：Dratman Ezequiel; Matera Guillermo^*

来源：Journal of Computational and Applied Mathematics, 2016, 292: 188-212.

DOI：10.1016/j.cam.2015.07.004

• Boundary-value problems
• Neumann boundary conditions
• Newton's method
• Newton-Mysovskikh's theorem
• Mesh-independence principle
• Complete cubic splines

We exhibit an algorithm which computes an 6-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 is an element of-approximation of the stationary solution is obtained. The algorithm performs roughly O((1/is an element of)(1/2)) flops and function evaluations.

• 出版日期2016-1-15