Homogenisation for Maxwell and Friends
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908600231788544 |
|---|---|
| author | Buchinger, Andreas Franz, Sebastian Skrepek, Nathanael Waurick, Marcus |
| author_facet | Buchinger, Andreas Franz, Sebastian Skrepek, Nathanael Waurick, Marcus |
| contents | We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homogenisation for Maxwell and Friends Buchinger, Andreas Franz, Sebastian Skrepek, Nathanael Waurick, Marcus Analysis of PDEs Numerical Analysis Mathematical Physics Functional Analysis Primary: 35B27, Secondary: 35Q61, 65M60, 65J08, 65J10 We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations. |
| title | Homogenisation for Maxwell and Friends |
| topic | Analysis of PDEs Numerical Analysis Mathematical Physics Functional Analysis Primary: 35B27, Secondary: 35Q61, 65M60, 65J08, 65J10 |
| url | https://arxiv.org/abs/2409.07084 |