Gaspard Kemlin (University of Stuttgart)

Title: Some recent work on the numerical analysis of two classes of nonlinear Schrödinger equations
Abstract: This talk will be divided in two parts and I will discuss some recent results on the numerical analysis of some classes of nonlinear Schrödinger equations. 

The first part of the talk will focus on Density functional theory (DFT): molecular simulation and electronic structure calculation are fundamental tools used in chemistry, condensed matter physics, molecular biology, materials science, nanoscience, etc. DFT is one of the most widely used methods today, offering a good compromise between efficiency and precision. It is a formidable problem, both mathematically and numerically, and requires a whole hierarchy of choices leading to a number of approximations and associated errors: choice of model, choice of discretisation basis, choice of solvers, truncation error, numerical error, etc. The aim of this first part is to present recent results dealing with these approximations. Firstly, I will introduce one of the most widely used models for electronic structure calculation. In concrete terms, this involves minimising a discrete energy on the Grassmann manifold, and I will then analyse the convergence of two classes of algorithms (direct minimisation methods and fixed point iterations) on this manifold. Then, we will see how the results established make it possible to obtain error bounds on quantities of practical interest (such as interatomic forces) or to improve the robustness of the calculation of response properties of materials. 

The second part of the talk will focus on the numerical simulation of the Gross-Pitaevskii (GP) equation via vortex tracking, as part of the SFB 1481 CRC. The GP equation plays a central role in various models of superfluids and condensed matter physics. A dominating feature is the occurrence of quantized vortices that effectively evolve according to a Hamiltonian system in the limit of point-like vortices. In this part, I will present an abstract mathematical model for which the analytic theory of the limiting Hamiltonian system is well known. I will show recent updates on the numerical simulation of this Hamiltonian system as well as first ideas to efficiently localize numerically the vortices in order to bring the analytic theory to the computational level.