arXiv subjects

Patrick Henning

Publications and source records attributed to Patrick Henning.

At least 19 recordsLinked to original sources

arXiv2026

Numerical vortex resolution for the Gross-Pitaevskii equation in the rapid rotation Thomas-Fermi scaling

In this paper we analyze finite element approximations of ground states of the Gross-Pitaevskii equation in the rapid rotation Thomas-Fermi scaling. In this regime, the healing length and vortex core size are of order $\eps \ll 1$, while the effective confinement potential may degenerate as the angular velocity approaches a critical value. In this setting, we analyze the $\eps$-dependence of the ground states and show that the local flatness of the energy landscape plays a decisive role for numerical resolution. More precisely, we establish mesh size conditions that guarantee the existence of discrete ground states in finite element spaces which are quasi-best approximations of an exact ground state. In particular, we prove that the absolute $H^1$-error behaves asymptotically like $h/\eps^2$. However, to enter this asymptotic regime, the mesh size must satisfy a significantly stronger resolution condition than the natural requirement $h \lesssim \eps$. The additional restriction is governed by the first spectral gap of the Riemannian Hessian of the energy functional at the ground state, which measures the local flatness of the energy surface. With this, our results provide an explanation of the mesh resolution required to capture vortex structures in rapidly rotating Bose-Einstein condensates and highlight the interplay between vortex core size, spectral stability, and discretization accuracy.

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arXiv2026

Structure and symmetry of the Gross-Pitaevskii ground-state manifold

The structure and degeneracy of ground states of the Gross-Pitaevskii energy functional play a central role in both analysis and computation, yet a precise characterization of the ground-state manifold in the presence of symmetries remains a fundamental challenge. In this paper, we establish sharp theoretical results describing the geometric structure of local minimizers and its implications for optimization algorithms. We show that when local minimizers are non-unique, the Morse-Bott condition provides a natural and sufficient criterion under which the ground-state set partitions into finitely many embedded submanifolds, each coinciding with an orbit generated by intrinsic symmetries of the energy functional, namely phase shifts and spatial rotations. This yields a structural characterization of the ground-state manifold in terms of these symmetries. Building on this insight, we characterize the local convergence behavior of general preconditioned Riemannian gradient methods (P-RG). Under the Morse-Bott condition, we derive sharp local $Q$-linear convergence estimates and prove that the condition holds if and only if the energy sequence generated by P-RG converges locally $Q$-linearly. In particular, on the ground-state set, the Morse-Bott condition is satisfied if and only if the minimizers decompose into finitely many symmetry orbits and P-RG exhibits local linear convergence nearby. When the condition fails, we establish a local sublinear convergence rate. Taken together, these results show that the Morse-Bott condition is the exact threshold separating linear from sublinear convergence, while determining the symmetry-induced structure of the ground-state manifold, connecting geometry, symmetry, and convergence behavior in a unified framework.

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arXiv2026

Nonlinear Inverse Iterations for Spin-Orbit Coupled Quantum Gases

This work concerns the computation of ground states of two-component spin-orbit coupled Bose-Einstein condensates (SO-coupled BECs), modelled by a coupled nonlinear eigenvalue problem of Gross-Pitaevskii type. Spin-orbit coupling gives rise to fascinating phenomena, including supersolid-like phases with spatially modulated densities. However, in such complex settings, conventional numerical approaches, such as generalized inverse iterations or gradient descent, often converge very slowly. To overcome this issue, we apply the concept of the J-method [E.~Jarlebring, S.~Kvaal, W.~Michiels. SIAM~J.~Sci.~Comput.~36-4,~2014] to construct a nonlinear inverse iteration scheme whose convergence can be accelerated through spectral shifting, analogous to techniques used for linear eigenproblems. For a fixed shift parameter, we establish local linear convergence rates determined by spectral gaps in the neighbourhood of each quasi-unique ground state. With adaptively chosen shifts, superlinear convergence is observed, which we verify through numerical experiments.

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arXiv2025

Metric-driven numerical methods

In this paper, we explore the concept of metric-driven numerical methods as a powerful tool for solving various types of multiscale partial differential equations. Our focus is on computing constrained minimizers of functionals - or, equivalently, by considering the associated Euler-Lagrange equations - the solution of a class of eigenvalue problems that may involve nonlinearities in the eigenfunctions. We introduce metric-driven methods for such problems via Riemannian gradient techniques, leveraging the idea that gradients can be represented in different metrics (so-called Sobolev gradients) to accelerate convergence. We show that the choice of metric not only leads to specific metric-driven iterative schemes, but also induces approximation spaces with enhanced properties, particularly in low-regularity regimes or when the solution exhibits heterogeneous multiscale features. In fact, we recover a well-known class of multiscale spaces based on the Localized Orthogonal Decomposition (LOD), now derived from a new perspective. Alongside a discussion of the metric-driven approach for a model problem, we also demonstrate its application to simulating the ground states of spin-orbit-coupled Bose-Einstein condensates.

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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 $\kappa$, capture the formation of lattices of quantized vortices. As the vortex cores shrink with increasing $\kappa$, while their number grows, it is essential to understand how $\kappa$ 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 $\kappa$-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

Convergence analysis of Sobolev Gradient flows for the rotating Gross-Pitaevskii energy functional

This paper studies the numerical approximation of the ground state of rotating Bose--Einstein condensates, formulated as the minimization of the Gross--Pitaevskii energy functional under a mass conservation constraint. To solve this problem, we consider three Sobolev gradient flow schemes: the $H_0^1$ scheme, the $a_0$ scheme, and the $a_u$ scheme. Convergence of these schemes in the non-rotating case was established by Chen et al., and the rotating $a_u$ scheme was analyzed in Henning et al. In this work, we prove the global convergence of the $H_0^1$ and $a_0$ schemes in the rotating case, and establish local linear convergence for all three schemes near the ground state. Numerical experiments confirm our theoretical findings.

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arXiv2025

The pollution effect for FEM approximations of the Ginzburg-Landau equation

In this paper, we investigate the approximation properties of solutions to the Ginzburg-Landau equation (GLE) in finite element spaces. Special attention is given to how the errors are influenced by coupling the mesh size $h$ and the polynomial degree $p$ of the finite element space to the size of the so-called Ginzburg-Landau material parameter $\kappa$. As observed in previous works, the finite element approximations to the GLE are suffering from a numerical pollution effect, that is, the best-approximation error in the finite element space converges under mild coupling conditions between $h$ and $\kappa$, whereas the actual finite element solutions possess poor accuracy in a large pre-asymptotic regime which depends on $\kappa$. In this paper, we provide a new error analysis that allows us to quantify the pre-asymptotic regime and the corresponding pollution effect in terms of explicit resolution conditions. In particular, we are able to prove that higher polynomial degrees reduce the pollution effect, i.e., the accuracy of the best-approximation is achieved under relaxed conditions for the mesh size. We provide both error estimates in the $H^1$- and the $L^2$-norm and we illustrate our findings with numerical examples.

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arXiv2025

Stable localized orthogonal decomposition in Raviart-Thomas spaces

This work proposes a computational multiscale method for the mixed formulation of a second-order linear elliptic equation subject to a homogeneous Neumann boundary condition, based on a stable localized orthogonal decomposition (LOD) in Raviart-Thomas finite element spaces. In the spirit of numerical homogenization, the construction provides low-dimensional coarse approximation spaces that incorporate fine-scale information from the heterogeneous coefficients by solving local patch problems on a fine mesh. The resulting numerical scheme is accompanied by a rigorous error analysis, and it is applicable beyond periodicity and scale-separation in spatial dimensions two and three. In particular, this novel realization circumvents the presence of pollution terms observed in a previous LOD construction for elliptic problems in mixed formulation. Finally, various numerical experiments are provided that demonstrate the performance of the method.

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arXiv2024

Riemannian conjugate Sobolev gradients and their application to compute ground states of BECs

This work considers the numerical computation of ground states of rotating Bose-Einstein condensates (BECs) which can exhibit a multiscale lattice of quantized vortices. This problem involves the minimization of an energy functional on a Riemannian manifold. For this we apply the framework of nonlinear conjugate gradient methods in combination with the paradigm of Sobolev gradients to investigate different metrics. Here we build on previous work that proposed to enhance the convergence of regular Riemannian gradients methods by an adaptively changing metric that is based on the current energy. In this work, we extend this approach to the branch of Riemannian conjugate gradient (CG) methods and investigate the arising schemes numerically. Special attention is given to the selection of the momentum parameter in search direction and how this affects the performance of the resulting schemes. As known from similar applications, we find that the choice of the momentum parameter plays a critical role, with certain parameters reducing the number of iterations required to achieve a specified tolerance by a significant factor. Besides the influence of the momentum parameters, we also investigate how the methods with adaptive metric compare to the corresponding realizations with a standard $H^1_0$-metric. As one of our main findings, the results of the numerical experiments show that the Riemannian CG method with the proposed adaptive metric along with a Polak-Ribi\'ere or Hestenes-Stiefel-type momentum parameter show the best performance and highest robustness compared to the other CG methods that were part of our numerical study.

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arXiv2024

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 $\kappa$. 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.

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arXiv2024

Convergence of a Riemannian gradient method for the Gross-Pitaevskii energy functional in a rotating frame

This paper investigates the numerical approximation of ground states of rotating Bose-Einstein condensates. This problem requires the minimization of the Gross-Pitaevskii energy $E$ on a Hilbert manifold $\mathbb{S}$. To find a corresponding minimizer $u$, we use a generalized Riemannian gradient method that is based on the concept of Sobolev gradients in combination with an adaptively changing metric on the manifold. By a suitable choice of the metric, global energy dissipation for the arising gradient method can be proved. The energy dissipation property in turn implies global convergence to the density $|u|^2$ of a critical point $u$ of $E$ on $\mathbb{S}$. Furthermore, we present a precise characterization of the local convergence rates in a neighborhood of each ground state $u$ and how these rates depend on the first spectral gap of $E^{\prime\prime}(u)$ restricted to the $L^2$-orthogonal complement of $u$. With this we establish the first convergence results for a Riemannian gradient method to minimize the Gross-Pitaevskii energy functional in a rotating frame. At the same, we refine previous results obtained in the case without rotation. The major complication in our new analysis is the missing isolation of minimizers, which are at most unique up to complex phase shifts. For that, we introduce an auxiliary iteration in the tangent space $T_{\mathrm{i} u} \mathbb{S}$ and apply the Ostrowski theorem to characterize the asymptotic convergence rates through a weighted eigenvalue problem. Afterwards, we link the auxiliary iteration to the original Riemannian gradient method and bound the spectrum of the weighted eigenvalue problem to obtain quantitative convergence rates. Our findings are validated in numerical experiments.

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arXiv2024

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 [D\"orich, 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 $\kappa$ and a stabilization parameter $\beta \ge 0$ that is used in the construction of the multiscale spaces. Furthermore, we analyze how $\kappa$ affects the necessary locality of the multiscale basis functions and we prove that the choice $\beta=0$ yields typically the highest accuracy. Our findings are supported by numerical experiments.

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arXiv2023

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

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 $\kappa$. 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 $\kappa$ into a mesh resolution condition, which can be done through error estimates that are explicit with respect to $\kappa$ 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 $\kappa$-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 $\kappa$ and $h$. Preasymptotic effects are observed for large mesh sizes $h$.

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arXiv2023

On discrete ground states of rotating Bose-Einstein condensates

The ground states of Bose-Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross-Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the $L^2$- and $H^1$-norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler-Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.

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arXiv2022

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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arXiv2022

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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arXiv2022

The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem

In this paper we consider the generalized inverse iteration for computing ground states of the Gross-Pitaevskii eigenvector problem (GPE). For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross-Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao, Q. Du, SIAM J. Sci. Comput., 25 (2004)] or the damped inverse iteration suggested in [P. Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]. Our analysis also reveals why the inverse iteration for the GPE does not react favourably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.

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arXiv2021

On optimal convergence rates for discrete minimizers of the Gross-Pitaevskii energy in LOD spaces

In this paper we revisit a two-level discretization based on the Localized Orthogonal Decomposition (LOD). It was originally proposed in [P.Henning, A.M{\aa}lqvist, D.Peterseim. SIAM J. Numer. Anal.52-4:1525-1550, 2014] to compute ground states of Bose-Einstein condensates by finding discrete minimizers of the Gross-Pitaevskii energy functional. The established convergence rates for the method appeared however suboptimal compared to numerical observations and a proof of optimal rates in this setting remained open. In this paper we shall close this gap by proving optimal order error estimates for the $L^2$- and $H^1$-error between the exact ground state and discrete minimizers, as well as error estimates for the ground state energy and the ground state eigenvalue. In particular, the achieved convergence rates for the energy and the eigenvalue are of $6$th order with respect to the mesh size on which the discrete LOD space is based, without making any additional regularity assumptions. These high rates justify the use of very coarse meshes, which significantly reduces the computational effort for finding accurate approximations of ground states. In addition, we include numerical experiments that confirm the optimality of the new theoretical convergence rates, both for smooth and discontinuous potentials.

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