Andreas Buchinger, Sebastian Franz, Nathanael Skrepek, Marcus Waurick
{"title":"麦克斯韦和朋友们的同质化","authors":"Andreas Buchinger, Sebastian Franz, Nathanael Skrepek, Marcus Waurick","doi":"arxiv-2409.07084","DOIUrl":null,"url":null,"abstract":"We refine the understanding of continuous dependence on coefficients of\nsolution operators under the nonlocal $H$-topology viz Schur topology in the\nsetting of evolutionary equations in the sense of Picard. We show that certain\ncomponents of the solution operators converge strongly. The weak convergence\nbehaviour known from homogenisation problems for ordinary differential\nequations is recovered on the other solution operator components. The results\nare underpinned by a rich class of examples that, in turn, are also treated\nnumerically, suggesting a certain sharpness of the theoretical findings.\nAnalytic treatment of an example that proves this sharpness is provided too.\nEven though all the considered examples contain local coefficients, the main\ntheorems and structural insights are of operator-theoretic nature and, thus,\nalso applicable to nonlocal coefficients. The main advantage of the problem\nclass considered is that they contain mixtures of type, potentially highly\noscillating between different types of PDEs; a prototype can be found in\nMaxwell's equations highly oscillating between the classical equations and\ncorresponding eddy current approximations.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Homogenisation for Maxwell and Friends\",\"authors\":\"Andreas Buchinger, Sebastian Franz, Nathanael Skrepek, Marcus Waurick\",\"doi\":\"arxiv-2409.07084\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We refine the understanding of continuous dependence on coefficients of\\nsolution operators under the nonlocal $H$-topology viz Schur topology in the\\nsetting of evolutionary equations in the sense of Picard. We show that certain\\ncomponents of the solution operators converge strongly. The weak convergence\\nbehaviour known from homogenisation problems for ordinary differential\\nequations is recovered on the other solution operator components. The results\\nare underpinned by a rich class of examples that, in turn, are also treated\\nnumerically, suggesting a certain sharpness of the theoretical findings.\\nAnalytic treatment of an example that proves this sharpness is provided too.\\nEven though all the considered examples contain local coefficients, the main\\ntheorems and structural insights are of operator-theoretic nature and, thus,\\nalso applicable to nonlocal coefficients. The main advantage of the problem\\nclass considered is that they contain mixtures of type, potentially highly\\noscillating between different types of PDEs; a prototype can be found in\\nMaxwell's equations highly oscillating between the classical equations and\\ncorresponding eddy current approximations.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"11 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.07084\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.07084","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.