TY - JOUR
T1 - Gradient Flow Finite Element Discretisations with Energy-Based Adaptivity for Excited States of Schrödinger's Equation
AU - Heid , Pascal
JO - Journal of Computational Mathematics
VL - 5
SP - 933
EP - 955
PY - 2023
DA - 2023/05
SN - 41
DO - http://doi.org/10.4208/jcm.2207-m2020-0302
UR - https://global-sci.org/intro/article_detail/jcm/21680.html
KW - Schrödinger's equation, Excited states, Gradient flows, Adaptive finite element methods.
AB - <p style="text-align: justify;">The purpose of this paper is to verify that the computational scheme from [Heid et al., <em>Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation</em>, J. Comput. Phys. <strong>436</strong>&nbsp;(2021)] for the numerical approximation of the ground state of the Gross-Pitaevskii equation can equally be applied for the effective approximation of excited states of Schrödinger&#39;s equation. That procedure employs an adaptive interplay of a Sobolev gradient flow iteration and a novel local mesh refinement strategy, and yields a guaranteed energy decay in each step of the algorithm. The computational tests in the present work highlight that this strategy is indeed able to approximate excited states, with (almost) optimal convergence rate with respect to the number of degrees of freedom.</p>