Vortex-capturing multiscale spaces for the Ginzburg-Landau equation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913663189778432 |
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| author | Blum, Maria Döding, Christian Henning, Patrick |
| author_facet | Blum, Maria Döding, Christian Henning, Patrick |
| contents | This paper considers minimizers of the Ginzburg-Landau energy functional in special multiscale spaces that are based on finite elements. The spaces are constructed by localized orthogonal decomposition techniques and their usage for solving the Ginzburg-Landau equation was first suggested in [Dörich, Henning, SINUM 2024]. In this work we further explore their approximation properties and give an analytical explanation for why vortex structures of energy minimizers can be captured more accurately in these spaces. We quantify the necessary mesh resolution in terms of the Ginzburg-Landau parameter $κ$ and a stabilization parameter $β\ge 0$ that is used in the construction of the multiscale spaces. Furthermore, we analyze how $κ$ affects the necessary locality of the multiscale basis functions and we prove that the choice $β=0$ yields typically the highest accuracy. Our findings are supported by numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14772 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vortex-capturing multiscale spaces for the Ginzburg-Landau equation Blum, Maria Döding, Christian Henning, Patrick Numerical Analysis 65N12, 65N15, 65N30, 35Q56 This paper considers minimizers of the Ginzburg-Landau energy functional in special multiscale spaces that are based on finite elements. The spaces are constructed by localized orthogonal decomposition techniques and their usage for solving the Ginzburg-Landau equation was first suggested in [Dörich, Henning, SINUM 2024]. In this work we further explore their approximation properties and give an analytical explanation for why vortex structures of energy minimizers can be captured more accurately in these spaces. We quantify the necessary mesh resolution in terms of the Ginzburg-Landau parameter $κ$ and a stabilization parameter $β\ge 0$ that is used in the construction of the multiscale spaces. Furthermore, we analyze how $κ$ affects the necessary locality of the multiscale basis functions and we prove that the choice $β=0$ yields typically the highest accuracy. Our findings are supported by numerical experiments. |
| title | Vortex-capturing multiscale spaces for the Ginzburg-Landau equation |
| topic | Numerical Analysis 65N12, 65N15, 65N30, 35Q56 |
| url | https://arxiv.org/abs/2405.14772 |