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Volume 41, Issue 5
Gradient Flow Finite Element Discretisations with Energy-Based Adaptivity for Excited States of Schrödinger's Equation

Pascal Heid

DOI: 10.4208/jcm.2207-m2020-0302

J. Comp. Math., 41 (2023), pp. 933-955.

Published online: 2023-05

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  • Abstract

The purpose of this paper is to verify that the computational scheme from [Heid et al., Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation, J. Comput. Phys. 436 (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'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.

  • Keywords

Schrödinger's equation, Excited states, Gradient flows, Adaptive finite element methods.

  • AMS Subject Headings

35Q40, 81Q05, 65N25, 65N30, 65N50

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

pascal.heid@maths.ox.ac.uk (Pascal Heid)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-933, author = {Heid , Pascal}, title = {Gradient Flow Finite Element Discretisations with Energy-Based Adaptivity for Excited States of Schrödinger's Equation}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {5}, pages = {933--955}, abstract = {

The purpose of this paper is to verify that the computational scheme from [Heid et al., Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation, J. Comput. Phys. 436 (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'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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2207-m2020-0302}, url = {http://global-sci.org/intro/article_detail/jcm/21680.html} }
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 -

The purpose of this paper is to verify that the computational scheme from [Heid et al., Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation, J. Comput. Phys. 436 (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'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.

Heid , Pascal. (2023). Gradient Flow Finite Element Discretisations with Energy-Based Adaptivity for Excited States of Schrödinger's Equation. Journal of Computational Mathematics. 41 (5). 933-955. doi:10.4208/jcm.2207-m2020-0302
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