2019/12/16 by Yvon Maday, Maday, Yvon, Carlo Marcati +1
Computer Science · Engineering · Mathematics · #35J10 #35P30 #65N12 #65N25 (Primary) #65N30 (Secondary) #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #Numerical methods in inverse problems #cs.NA #math.AP #math.NA #msc:35J10 #msc:35P30 #msc:65N12 #msc:65N25 #msc:65N30
paper · pdf · doi:10.48550/arxiv.1912.07483
openalex publication_date 2019/12/16 · arxiv created 2022/10/24 · arxiv updated 2022/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined hp discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined hp space the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.