On numerical methods for nonlinear singularly perturbed Schrödinger problems

In this article, we develop a new method to compute positive mountain pass solutions, which improves the results for a large range of singular parameters. We extend ideas from MPA considering the singulary perturbed problems by developing a finite element approach mixed with steepest descend directions. We use a modified line search method based on Armijo's rule for improving the Newton search and Patankar trick for preserving the positiveness of the solution. To improve the range of the singular parameter, adaptive methods based on Dual Weighted Residual method are used. Our numerical experiments are performed with the deal.II library and we show that it is possible to get solutions for ε=10−6 improving the current results in four orders of magnitude. At this level, machine precision must be considered for further studies.