Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation

TL;DR

This paper develops and analyzes fully discrete Galerkin schemes for the nonlinear, nonlocal Hartree equation, providing theoretical guarantees for existence, uniqueness, regularity, and approximation in various discretization settings.

Contribution

It introduces novel fully discrete Galerkin schemes for the Hartree equation and proves their mathematical properties, enabling accurate numerical simulations of complex nonlocal quantum dynamics.

Findings

Proved existence and uniqueness of solutions for the schemes

Established regularity and approximation properties

Set the foundation for controlled numerical computations

Abstract

We study the time dependent Hartree equation in the continuum, the semidiscrete, and the fully discrete setting. We prove existence-uniqueness, regularity, and approximation properties for the respective schemes, and set the stage for a controlled numerical computation of delicate nonlinear and nonlocal features of the Hartree dynamics in various physical applications.