Error bounds for discrete minimizers of the Ginzburg-Landau energy in the high-$κ$ regime

TL;DR

This work analyzes discrete minimizers of the Ginzburg–Landau energy in finite element spaces in the high-$κ$ regime, where vortices demand fine meshes. It develops a κ-weighted variational framework, proves existence and stability of discrete minimizers, and derives error bounds that are explicit in both $κ$ and the mesh width $h$, including convergence rates for linear Lagrange elements and extensions to Localized Orthogonal Decomposition (LOD) spaces. Numerical experiments confirm the predicted κ-scaling (e.g., $|u-u_h|_{H^1_κ} \sim κ^2 h$, $|u-u_h|_{L^2} \sim κ^2 h^2$) and reveal preasymptotic effects for large $h$. The results provide practical guidance on mesh resolution relative to $κ$ and illustrate the potential of LOD to improve performance in the high-$κ$ regime, with robust a priori error bounds and verifiable local uniqueness up to gauge.

Abstract

In this work, we study discrete minimizers of the Ginzburg-Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg-Landau parameter $κ$. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of $κ$ into a mesh resolution condition, which can be done through error estimates that are explicit with respect to $κ$ and the spatial mesh width $h$. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted $κ$-weighted norm. Afterwards we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we further explore the asymptotic optimality of our derived $L^2$- and $H^1$-error estimates with respect to $κ$ and $h$. Preasymptotic effects are observed for large mesh sizes $h$.

Figures (5)

• Figure 1: For a given minimizer $u$, the figure illustrates the circle line parametrized by the $2\pi$-periodic curve $\gamma : t\mapsto \exp(- \mathrm{i} t) u$ for $t \in [-\pi ,\pi)$. The tangent direction in $u$ is given by $\gamma^{\prime}(0) = \mathrm{i} u$ and the energy $E$ is constant and minimal on the whole circle line, i.e. $\frac{\hbox{\rm d}^2}{\hbox{\rm d}t^2} E(\space\gamma(t)\space)= 0$.
• Figure 2: Convergence in the mesh size $h$ for $\kappa$-weighted errors in the $H^1_\kappa$- and $L^2$-norm and for the energy, for $\kappa = 8,10,17,24$. The errors between $u$ and $u_{h}$ in $L^2$ and $H^1_\kappa$ are scaled by $\kappa^{-2}$ and the error in energy by $\kappa^{-4}$. The dotted lines indicate the corresponding errors (in $L^2$ and $H^1_\kappa$ respectively) between $u$ and its best-approximation $\textup{R}_{\kappa,h}(u)$ in $V_{h}$ with respect to $\hat{a}_\kappa(\cdot,\cdot)$, cf. \ref{['best-approxi-H1kappa']}. The dashed lines indicate order $\mathcal{O}(h)$ in the upper left figure, and order $\mathcal{O}(h^2)$ in the upper left and bottom right figure.
• Figure 3: Boundedness of the scaled $H^1_\kappa$-norm and energy $E$ with respect to $h$ for $\kappa = 8,10,17,24$. The dotted lines show the energy of the best-approximation $\textup{R}_{\kappa,h}(u)$ in $V_{h}$ with respect to $\hat{a}_\kappa(\cdot,\cdot)$.
• Figure 4: Minimizers for the Ginzburg--Landau parameter $\kappa =20$ and different mesh widths $h \approx 8\cdot 10^{-2}, 4\cdot 10^{-2}, 2\cdot 10^{-2}, 1 \cdot 10^{-2}$ (from left to right).
• Figure 5: Different minimizers corresponding to the Ginzburg--Landau parameters $\kappa = 8,10,17,24$ (from left to right) for $h \approx 2.5 \cdot 10^{-3}$.

Theorems & Definitions (54)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4: Local uniqueness up to gauge transformation
• Proposition 2.6
• proof
• Remark 2.7: Verifying the local uniqueness numerically