Vortex-capturing multiscale spaces for the Ginzburg-Landau equation

TL;DR

The paper demonstrates that vortex-rich minimizers of the Ginzburg–Landau energy can be efficiently captured on multiscale, localized orthogonal decomposition spaces, reducing the mesh resolution required in the high-$\kappa$ regime. It develops a stabilized, $\kappa$-aware framework with a $β$-dependent bilinear form and constructs both ideal and localized LOD spaces via element correctors whose decay is exponential in the localization depth. Theoretical results show a third-order $O(h^3)$ convergence in the ideal setting, with fully localized versions preserving these rates up to a controllable localization error, and numerical experiments confirm improved vortex resolution and accuracy compared to standard FE, especially for $β\approx 0$. The work provides practical guidance on choosing $β$ and the localization parameter $ℓ$, and demonstrates that LOD-based solvers can achieve accurate vortex lattices on coarse meshes, offering substantial efficiency gains for simulating high-$\kappa$ superconductors.

Abstract

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.

Figures (7)

• Figure 1: Patch neighborhoods of a triangle $T$: $T={\color{dimgray}\bullet}$, $\textup{N}(T)={\color{dimgray}\bullet} \cup {\color{darkgray}\bullet}$ and $\textup{N}(\textup{N}(T))={\color{dimgray}\bullet} \cup {\color{darkgray}\bullet} \cup {\color{lightgray}\bullet}$
• Figure 2: Vortex patterns of the reference solution $|u_{\text{\tiny ref}}^{{\text{\normalfont\tiny FEM}}}|$ with $\kappa=8,12,16,20,32$.
• Figure 3: Convergence of the $H^1_\kappa$-errors $\varepsilon_h^{{\text{\normalfont\tiny LOD}}}$ (left) and $\kappa^{-3} \varepsilon_h^{{\text{\normalfont\tiny LOD}}}$ (right) with $\beta=1$.
• Figure 4: Comparison of $\varepsilon_h^{{\text{\normalfont\tiny LOD}}}$ (solid lines) and the $H^1_\kappa$-best-approximation error $\varepsilon_h^{\mathrm{best}}$ from \ref{['best-approx-error-Vlodell']} (dashed lines) with $\beta=1$.
• Figure 5: Left: Convergence of the $H^1_\kappa$-error $\varepsilon_h^{{\text{\normalfont\tiny LOD}}}$ with $\beta=0$. Right: Comparison of the $H^1_\kappa$-error $\varepsilon_h^{{\text{\normalfont\tiny LOD}}}$ for the stabilization parameters $\beta=1$ vs. $\beta=0$ with $\kappa = 8,12,16$.

Theorems & Definitions (33)

• Theorem 2.1
• Proposition 2.2
• proof : Proof of Proposition \ref{['coercivity_secE_u']}
• Remark 2.3: Boundary condition
• Lemma 2.4
• proof : Proof of Lemma \ref{['Lemma_2.1']}
• Theorem 3.1
• Lemma 4.1
• proof
• Definition 4.2: $a_{\beta}(\cdot,\cdot)$-based LOD space