Gradient Flow Finite Element Discretisations with Energy-Based $hp$-Adaptivity for the Gross-Pitaevskii Equation with Angular Momentum
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911324566454272 |
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| author | Heid, Pascal Houston, Paul Stamm, Benjamin Wihler, Thomas P. |
| author_facet | Heid, Pascal Houston, Paul Stamm, Benjamin Wihler, Thomas P. |
| contents | This article deals with the stationary Gross-Pitaevskii non-linear eigenvalue problem in the presence of a rotating magnetic field that is used to model macroscopic quantum effects such as Bose-Einstein condensates (BECs). In this regime, the ground-state wave-function can exhibit an a priori unknown number of quantum vortices at unknown locations, which necessitates the exploitation of adaptive numerical strategies. To this end, we consider the conforming finite element method in combination with a discrete Sobolev gradient descent, which is guided by the energy-topology of the problem, to address the nonlinearity. In addition, a key novelty of this work is an $hp$-adaptive strategy that is solely based on energy decay rather than a posteriori error estimators for the refinement process. Numerical results demonstrate that the $hp$-adaptive strategy is highly efficient in terms of accuracy to compute the ground-state wave function and energy for several test problems where we observe exponential convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17680 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient Flow Finite Element Discretisations with Energy-Based $hp$-Adaptivity for the Gross-Pitaevskii Equation with Angular Momentum Heid, Pascal Houston, Paul Stamm, Benjamin Wihler, Thomas P. Numerical Analysis This article deals with the stationary Gross-Pitaevskii non-linear eigenvalue problem in the presence of a rotating magnetic field that is used to model macroscopic quantum effects such as Bose-Einstein condensates (BECs). In this regime, the ground-state wave-function can exhibit an a priori unknown number of quantum vortices at unknown locations, which necessitates the exploitation of adaptive numerical strategies. To this end, we consider the conforming finite element method in combination with a discrete Sobolev gradient descent, which is guided by the energy-topology of the problem, to address the nonlinearity. In addition, a key novelty of this work is an $hp$-adaptive strategy that is solely based on energy decay rather than a posteriori error estimators for the refinement process. Numerical results demonstrate that the $hp$-adaptive strategy is highly efficient in terms of accuracy to compute the ground-state wave function and energy for several test problems where we observe exponential convergence. |
| title | Gradient Flow Finite Element Discretisations with Energy-Based $hp$-Adaptivity for the Gross-Pitaevskii Equation with Angular Momentum |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2412.17680 |