Christian Döding, Benjamin Dörich, Patrick Henning
{"title":"A multiscale approach to the stationary Ginzburg-Landau equations of superconductivity","authors":"Christian Döding, Benjamin Dörich, Patrick Henning","doi":"arxiv-2409.12023","DOIUrl":null,"url":null,"abstract":"In this work, we study the numerical approximation of minimizers of the\nGinzburg-Landau free energy, a common model to describe the behavior of\nsuperconductors under magnetic fields. The unknowns are the order parameter,\nwhich characterizes the density of superconducting charge carriers, and the\nmagnetic vector potential, which allows to deduce the magnetic field that\npenetrates the superconductor. Physically important and numerically challenging\nare especially settings which involve lattices of quantized vortices which can\nbe formed in materials with a large Ginzburg-Landau parameter $\\kappa$. In\nparticular, $\\kappa$ introduces a severe mesh resolution condition for\nnumerical approximations. In order to reduce these computational restrictions,\nwe investigate a particular discretization which is based on mixed meshes where\nwe apply a Lagrange finite element approach for the vector potential and a\nlocalized orthogonal decomposition (LOD) approach for the order parameter. We\njustify the proposed method by a rigorous a-priori error analysis (in $L^2$ and\n$H^1$) in which we keep track of the influence of $\\kappa$ in all error\ncontributions. This allows us to conclude $\\kappa$-dependent resolution\nconditions for the various meshes and which only impose moderate practical\nconstraints compared to a conventional finite element discretization. Finally,\nour theoretical findings are illustrated by numerical experiments.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","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-2409.12023","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this work, we study the numerical approximation of minimizers of the
Ginzburg-Landau free energy, a common model to describe the behavior of
superconductors under magnetic fields. The unknowns are the order parameter,
which characterizes the density of superconducting charge carriers, and the
magnetic vector potential, which allows to deduce the magnetic field that
penetrates the superconductor. Physically important and numerically challenging
are especially settings which involve lattices of quantized vortices which can
be formed in materials with a large Ginzburg-Landau parameter $\kappa$. In
particular, $\kappa$ introduces a severe mesh resolution condition for
numerical approximations. In order to reduce these computational restrictions,
we investigate a particular discretization which is based on mixed meshes where
we apply a Lagrange finite element approach for the vector potential and a
localized orthogonal decomposition (LOD) approach for the order parameter. We
justify the proposed method by a rigorous a-priori error analysis (in $L^2$ and
$H^1$) in which we keep track of the influence of $\kappa$ in all error
contributions. This allows us to conclude $\kappa$-dependent resolution
conditions for the various meshes and which only impose moderate practical
constraints compared to a conventional finite element discretization. Finally,
our theoretical findings are illustrated by numerical experiments.