# Gradient Flow Finite Element Discretizations with Energy-Based Adaptivity for the Gross-Pitaevskii Equation

## TL;DR

This paper introduces an energy-based adaptive finite element method for solving the steady-state Gross-Pitaevskii equation, combining gradient flow iterations with mesh refinement to achieve high accuracy and optimal convergence.

## Contribution

The paper develops a novel adaptive finite element approach based on energy minimization and gradient flow for the Gross-Pitaevskii equation, demonstrating improved accuracy and convergence.

## Key findings

- Achieves highly accurate solutions for the Gross-Pitaevskii equation.
- Demonstrates optimal convergence rates with respect to degrees of freedom.
- Effective combination of gradient flow and adaptive mesh refinement.

## Abstract

We present an effective adaptive procedure for the numerical approximation of the steady-state Gross-Pitaevskii equation. Our approach is solely based on energy minimization, and consists of a combination of gradient flow iterations and adaptive finite element mesh refinements. Numerical tests show that this strategy is able to provide highly accurate results, with optimal convergence rates with respect to the number of freedom.