Gradient flow finite element discretisations with energy-based adaptivity for excited states of Schr\"odingers equation

TL;DR

This paper introduces an adaptive finite element method combining gradient flow and mesh refinement to accurately compute excited states of Schrödinger's equation, ensuring energy decay and optimal convergence.

Contribution

It develops a novel adaptive scheme that guarantees energy decay and achieves optimal convergence for excited state computations in Schrödinger's equation.

Findings

Achieves highly accurate excited state approximations.

Demonstrates optimal convergence rate with respect to degrees of freedom.

Ensures guaranteed energy decay at each iteration.

Abstract

We present an effective numerical procedure, which is based on the computational scheme from [Heid et al., arXiv:1906.06954], for the numerical approximation of excited states of Schr\"odingers equation. In particular, this procedure employs an adaptive interplay of gradient flow iterations and local mesh refinements, leading to a guaranteed energy decay in each step of the algorithm. The computational tests highlight that this strategy is able to provide highly accurate results, with optimal convergence rate with respect to the number of degrees of freedom.