arXiv subjects

Christian Döding

Publications and source records attributed to Christian Döding.

12 recordsLinked to original sources

arXiv2026

On second-order optimality in the high-$κ$ regime of the Ginzburg-Landau model

We study energy minimizers of the Ginzburg-Landau (GL) free energy, a fundamental model of superconductivity. We address the high-$κ$ regime, the regime of a large GL parameter, in which energy minimizers exhibit vortex structures whose finite element approximations require a fine mesh resolution. This difficulty is reflected in the error analysis of discrete minimizers, which relies on a second-order optimality condition. The spectrum of the energy's second Fréchet derivative must be bounded away from zero up to symmetry. In practice, the associated spectral gap decreases rapidly with the GL parameter. This degrades the quality of the approximations because the GL parameter directly enters as an additional factor in the error estimates. Although a polynomial dependence of the spectral gap on the GL parameter has been conjectured, its precise behavior remains unclear. As a first step toward addressing this issue, we compute the spectral gap based on a finite element approximation for a range of GL parameters, providing numerical evidence for the conjectured polynomial dependence.

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arXiv2026

Optimization-Based Identification of Effective Coefficients for Wave Equations in Spatio-Temporal Metamaterials

We study the identification of effective coefficients for wave equations in heterogeneous media. Such equations arise in the modeling of spatio-temporal metamaterials, where the underlying material properties exhibit variations in both space and time. While homogenization provides effective models in the asymptotic regime of vanishing microscopic scales, determining macroscopic parameters from observations of wave propagation remains challenging. We introduce an optimization-based method that identifies a constant effective coefficient by minimizing a cost functional. The approach is designed for situations in which the underlying space-time-dependent coefficient is unknown, while the solution is available in space and time by measurements. We extend optimization-based coefficient identification techniques from elliptic multiscale problems to wave equations and prove that, provided a homogenized limit exists, the identified coefficient converges to the homogenized coefficient as the microscopic scale tends to zero. Furthermore, we establish convergence of the corresponding effective solution towards the heterogeneous solution, including strong convergence in $L^2$ and weak convergence of first-order space and time derivatives. Numerical experiments further demonstrate the performance of the method for non-periodic space-time heterogeneous media, including cases for which a homogenized limit is not known to exist.

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arXiv2026

GLENN: Neural network-enhanced computation of Ginzburg-Landau energy minimizers

In this work, we propose a neural network-enhanced finite element strategy to compute the minimizer of the Ginzburg-Landau energy based on an unsupervised deep Ritz-type strategy. We treat the parameter $κ$ as a variable input parameter to obtain possible minimizers for a large range of $κ$-values. This allows for two possible strategies: 1) The neural network may be extensively trained to work as a stand-alone solver. 2) Neural network results are used as starting values for a subsequent classical iterative minimization procedure. The latter strategy particularly circumvents the missing reliability of the neural network-based approach. Numerical examples are presented that show the potential of the proposed strategy.

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arXiv2026

Local and nonlocal homogenization of wave propagation in time-varying media

Temporal metamaterials are artificially manufactured materials with time-dependent material properties that exhibit interesting phenomena when waves propagate through them. The propagation of electromagnetic waves in such time-varying dielectric media is governed by Maxwell's equations, which lead to wave equations with temporal highly oscillatory coefficients for the electric and magnetic fields. In this study, we analyze the effective behavior of electromagnetic fields in time-varying metamaterials using a formal two-scale asymptotic expansion. We provide a mathematical derivation of the effective equations for the leading-order homogenized solution, as well as for the first- and second-order corrections of the effective solution. While the effective solution and the first-order correction are governed by local material laws, we reveal a nonlocal constitutive relation for the second-order corrections. Special attention is also paid to temporal interface conditions through initial values of the homogenized equations. The results provide a mathematically justified framework for the effective description of wave-type equations of time-varying media, applicable to models in optics, elasticity, and acoustics.

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arXiv2026

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

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 $κ$. In particular, $κ$ 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 $L^2$ and $H^1$) in which we keep track of the influence of $κ$ in all error contributions. This allows us to conclude $κ$-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.

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arXiv2025

The Ginzburg-Landau equations: Vortex states and numerical multiscale approximations

In this review article, we provide an overview of recent advances in the numerical approximation of minimizers of the Ginzburg-Landau energy in multiscale spaces. Such minimizers represent the most stable states of type-II superconductors and, for large material parameters $κ$, capture the formation of lattices of quantized vortices. As the vortex cores shrink with increasing $κ$, while their number grows, it is essential to understand how $κ$ should couple to the mesh size in order to correctly resolve the vortex patterns in numerical simulations. We summarize and discuss recent developments based on LOD (Localized Orthogonal Decomposition) multiscale methods and review the corresponding error estimates that explicitly reflect the $κ$-dependence and the observed superconvergence. In addition, we include several minor refinements and extensions of existing results by incorporating techniques from recent contributions to the field. Finally, numerical experiments are presented to illustrate and support the theoretical findings.

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arXiv2025

Vortex-capturing multiscale spaces for the Ginzburg-Landau equation

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.

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arXiv2024

Localized Orthogonal Decomposition Methods vs. Classical FEM for the Gross-Pitaevskii Equation

The time-dependent Gross-Pitaevksii equation (GPE) is a nonlinear Schrödinger equation which is used in quantum physics to model the dynamics of Bose-Einstein condensates. In this work we consider numerical approximations of the GPE based on a multiscale approach known as the localized orthogonal decomposition. Combined with an energy preserving time integrator one derives a method which is of high order in space and time under mild regularity assumptions. In previous work, the method has been shown to be numerically very efficient compared to first order Lagrange FEM. In this paper, we further investigate the performance of the method and compare it with higher order Lagrange FEM. For rough problems we observe that the novel method performs very efficient and retains its high order, while the classical methods can only compete well for smooth problems.

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arXiv2024

A two level approach for simulating Bose-Einstein condensates by localized orthogonal decomposition

In this work, we consider the numerical computation of ground states and dynamics of single-component Bose-Einstein condensates (BECs). The corresponding models are spatially discretized with a multiscale finite element approach known as Localized Orthogonal Decomposition (LOD). Despite the outstanding approximation properties of such a discretization in the context of BECs, taking full advantage of it without creating severe computational bottlenecks can be tricky. In this paper, we therefore present two fully-discrete numerical approaches that are formulated in such a way that they take special account of the structure of the LOD spaces. One approach is devoted to the computation of ground states and another one for the computation of dynamics. A central focus of this paper is also the discussion of implementation aspects that are very important for the practical realization of the methods. In particular, we discuss the use of suitable data structures that keep the memory costs economical. The paper concludes with various numerical experiments in 1d, 2d and 3d that investigate convergence rates and approximation properties of the methods and which demonstrate their performance and computational efficiency, also in comparison to spectral and standard finite element approaches.

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arXiv2024

Algebraic rates of stability for front-type modulated waves in Ginzburg Landau equations

We consider the stability of front-type modulated waves in the complex Ginzburg-Landau equation (CGL). The waves occur in the bistable regime (e.g. of the quintic CGL) and connect the zero state to a spatially homogenous state oscillating in time. For initial perturbations that decay at a certain algebraic rate, we prove convergence to the wave with asymptotic phase. The convergence holds in algebraically weighted Sobolev norms and with an algebraic rate in time, where the asymptotic phase is approached by one order less than the profile. On the technical side we use the theory of exponential trichotomies to separate the spatial modes into growing, weakly decaying, and strongly decaying ones. This allows us to derive resolvent and semigroup estimates in weighted Sobolev norms and to close the argument with a Gronwall lemma involving algebraic weights.

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arXiv2023

Uniform $L^\infty$-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation

In this paper, we consider an energy-conserving continuous Galerkin discretization of the Gross-Pitaevskii equation with a magnetic trapping potential and a stirring potential for angular momentum rotation. The discretization is based on finite elements in space and time and allows for arbitrary polynomial orders. It was first analyzed in [O. Karakashian, C. Makridakis; SIAM J. Numer. Anal. 36(6):1779-1807, 1999] in the absence of potential terms and corresponding a priori error estimates were derived in 2D. In this work we revisit the approach in the generalized setting of the Gross-Pitaevskii equation with rotation and we prove uniform $L^\infty$-bounds for the corresponding numerical approximations in 2D and 3D without coupling conditions between the spatial mesh size and the time step size. With this result at hand, we are in particular able to extend the previous error estimates to the 3D setting while avoiding artificial CFL conditions.

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arXiv2021

Stability of Traveling Oscillating Fronts in Complex Ginzburg Landau Equations

Traveling oscillating fronts (TOFs) are specific waves of the form $U_\star (x,t) = e^{-i ωt} V_\star(x - ct)$ with a profile $V_{\star}$ which decays at $- \infty$ but approaches a nonzero limit at $+\infty$. TOFs usually appear in complex Ginzburg Landau equations of the type $U_t = αU_{xx} + G(|U|^2)U$. In this paper we prove a theorem on the asymptotic stability of TOFs, where we allow the initial perturbation to be the sum of an exponentially localized part and a front-like part which approaches a small but nonzero limit at $+ \infty$. The underlying assumptions guarantee that the operator, obtained from linearizing about the TOF in a co-moving and co-rotating frame, has essential spectrum touching the imaginary axis in a quadratic fashion and that further isolated eigenvalues are bounded away from the imaginary axis. The basic idea of the proof is to consider the problem in an extended phase space which couples the wave dynamics on the real line to the ODE dynamics at $+ \infty$. Using slowly decaying exponential weights, the framework allows to derive appropriate resolvent estimates, semigroup techniques, and Gronwall estimates.

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