<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>. In particular, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> introduces a severe mesh resolution condition for numerical approximations. In order to reduce these computational restrictions, we investigate a particular discretization which is based on mixed meshes where we apply a Lagrange finite element approach for the vector potential and a localized orthogonal decomposition (LOD) approach for the order parameter. We justify the proposed method by a rigorous a-priori error analysis (in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>) in which we keep track of the influence of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> in the main error contributions. This allows us to conclude <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>-dependent resolution conditions for the various meshes and which only impose moderate practical constraints compared to a conventional finite element discretization. While our results only provide information on the approximability of the minimizers, we conclude by further proposing a minimization procedure to illustrate our theoretical findings by numerical experiments.</p>

A multiscale approach to the stationary Ginzburg–Landau equations of superconductivity

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2026年5月16日

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Christian Döding,
Benjamin Dörich,
Patrick Henning

摘要

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 $ $κ$ . In particular, $\kappa $ $κ$ 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 $L^2$ $L^{2}$ and $H^1$ $H^{1}$ ) in which we keep track of the influence of $\kappa $ $κ$ in the main error contributions. This allows us to conclude $\kappa $ $κ$ -dependent resolution conditions for the various meshes and which only impose moderate practical constraints compared to a conventional finite element discretization. While our results only provide information on the approximability of the minimizers, we conclude by further proposing a minimization procedure to illustrate our theoretical findings by numerical experiments.