Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems

Abstract

A. We study a class of nonlinear eigenvalue problems of Scrödinger 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 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. In addition, we investigate the role of pointwise convergence in the doubling of the convergence rate for the eigenvalues with respect to the convergence rate of eigenfunctions. We conclude with a series of numerical tests to validate the theoretical results.

Variational projector-augmented wave method: a full-potential approach for electronic structure calculations in solid-state physics

“…It has been applied successfully to characterize precisely the behaviour of the electronic wave function close the nucleus [14,16] and used in the analysis of the muffin-tin and LAPW methods [9]. It is also a key element of the analysis of the convergence of hp-finite elements approximation for similar models [26,27]. The interested reader may refer to [20,13] for a detailed exposition of this theory.…”

Weighted analyticity of Hartree-Fock eigenfunctions

“…The results presented in this paper rely heavily on the theory of Kondrat'ev-type weighted Sobolev spaces, that we introduce here. We also recall-mostly from [MM19a], for self-containednessa series of technical results that are ultimately necessary for the proof of Theorem 1.…”

“…Analytic regularity of solutions to linear elliptic systems in polygons and polyhedra has been analyzed, e.g, in [GS06,CDN12]. Concerning nonlinear problems, we mention our work on nonlinear Schrödinger equations [MM19a] and on the Navier-Stokes equation in plane polygons [MS20]. For a general theory of elliptic regularity in weighted spaces, we refer the reader to, e.g., [Gri85,KMR97,KMR01,MR10], and the recent work [DHSS19].…”

“…The weighted analytic regularity of the solutions to Hartree-Fock and more general elliptic problems has important and far-reaching consequences for the numerical solution of those problems. We can, indeed, obtain exponential rates of convergence of solutions obtained via numerical methods based on finite elements, see [SSW13b,SSW13a] for a general approximation theory and [MM19b,MM19a,HSW19] for applications to linear and nonlinear eigenvalue problems, and on virtual elements [ČGM + 20]. In addition, nonlinear approximation techniques based on tensor compression and on the solution of partial differential equation in tensor-formatted form also provide exponentially convergent solutions to problems with weighted analytic solutions [MRS19].…”

Tensor Rank bounds for Point Singularities in $\mathbb{R}^3$

“…Solutions to scalar elliptic problems with constant coefficients belong to analytictype weighted spaces [12, 13], as do the flow and pressure obtained with the Stokes [28] and Navier-Stokes [54] equations in polygons. Furthermore, eigenfunctions to three-dimensional linear [51] and nonlinear [50] Schrödinger equations are weighted analytic. In quantum chemistry, the wave functions computed with the non-relativistic Hartree-Fock models for electronic structure calculations are also analytic in weighted Sobolev spaces [52,Section 7.4], [9], with point singularities at the nuclei.…”