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Paola Pozzi

Publications and source records attributed to Paola Pozzi.

17 recordsLinked to original sources
arXiv2025

Minimal horizontal triods: Analysis and computation

In this article we investigate the question of finding a network configuration of minimal length connecting three given points in the Heisenberg group. After proving existence of (possibly degenerate) minimal horizontal triods, we investigate their characterization. We then formulate a horizontal curve shortening flow that deforms any given suitable initial triod into a critical point for the length functional. Numerical experiments based on a stable fully discrete finite element scheme provide useful insights into the rich landscape of this sub-Riemannian geometry.

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arXiv2025

A surface finite element scheme for a stochastic PDE on an evolving curve

In this paper we consider an ESFEM method for the advection and diffusion of a scalar quantity on a moving closed curve. The diffusion process is controlled by a forcing term that may include a rough term (specifically a stochastic noise) which in particular destroys the classical time differentiability properties of the solution. We provide a suitable variational solution concept and a fully discrete FEM discretization. Our error analysis appropriately generalizes classical estimates to this weaker setting. We present some numerical simulations that confirm our theoretical findings.

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arXiv2023

Stability analysis for the anisotropic curve shortening flow of planar networks

In this article we study the anisotropic curve shortening flow for a planar network of three curves with fixed endpoints and which meet in a triple junction. We show that the anisotropic curvature energy fulfills a Lojasiewicz-Simon gradient inequality and use this knowledge to derive stability results for the flow. Precisely, in our main theorem we show that for any initial data, which are $C^{2,α}$-close to a (local) energy minimizer, the flow exists globally and converges to a possibly different energy minimum.

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arXiv2022

Convergence of a scheme for elastic flow with tangential mesh movement

Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy for this purpose. The approach effectively also penalizes the length of the curve, and equilibrium shapes are equivalent to stationary points of the elastic energy augmented with the length functional. Our numerical method is based on linear parametric finite elements. Following the lines of K Deckelnick, and G Dziuk (Math Comp 78, 266 (2009), 645-671) we prove convergence and establish error estimates, noting that the addition of the Dirichlet energy simplifies the analysis in comparison with the length functional. We also present a simple semi-implicit time discretization and discuss some numerical result that support the theory.

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arXiv2020

Anisotropic curvature flow of immersed networks

We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $L^2$ norm of the anisotropic curvature blows up.

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arXiv2020

Elastic flow of networks: short-time existence result

In this paper we study the $L^2$-gradient flow of the penalized elastic energy on networks of $q$-curves in $\R^{n}$ for $q \geq 3$. Each curve is fixed at one end-point and at the other is joint to the other curves at a movable $q$-junction. For this geometric evolution problem with natural boundary condition we show the existence of smooth solutions for a (possibly) short interval of time. Since the geometric problem is not well-posed, due to the freedom in reparametrization of curves, we consider a fourth-order non-degenerate parabolic quasilinear system, called the analytic problem, and show first a short-time existence result for this parabolic system. The proof relies on applying Solonnikov's theory on linear parabolic systems and Banach fixed point theorem in proper Hölder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.

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arXiv2019

A converging finite element scheme for motion by curvature of a network with a triple junction

A new semi-discrete finite element scheme for the evolution of three parametrized curves by curvature flow that are connected by a triple junction is presented and analyzed. In this triple junction, conditions are imposed on the angles at which the curves meet. One of the key problems in analyzing motion of networks by curvature law is the choice of a tangential velocity that allows for motion of the triple junction, does not lead to mesh degeneration, and is amenable to an error analysis. Our approach consists in considering a perturbation of a classical smooth formulation. The problem we propose admits a natural variational formulation that can be discretized with finite elements. The perturbation can be made arbitrarily small when a regularization parameter shrinks to zero. Convergence of the new scheme including optimal error estimates are proved. These results are supported by some numerical tests. Finally, the influence of the small regularization parameter on the properties of scheme and the accuracy of the results is numerically investigated.

math.NA↗
arXiv2019

A second order gradient flow of p-elastic planar networks

We consider a second order gradient flow of the p-elastic energy for a planar theta-network of three curves with fixed lengths. We construct a weak solution of the flow by means of an implicit variational scheme. We show long-time existence of the evolution and convergence to a critical point of the energy.

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arXiv2018

Flow of elastic networks: long-time existence result

We provide a long-time existence and sub-convergence result for the elastic flow of a three network in $\mathbb{R}^{n}$ under some mild topological assumptions. The evolution is such that the sum of the elastic energies of the three curves plus their weighted lengths decrease in time. Natural boundary conditions are considered at the boundary of the curves and at the triple junction.

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arXiv2018

A gradient flow for the p-elastic energy defined on closed planar curves

We study the evolution of closed inextensible planar curves under a second order flow that decreases the $p$-elastic energy. A short time existence result for $p \in (1,\infty)$ is obtained via a minimizing movements method. For $p = 2$, that is in the case of the classic elastic energy, long-time existence is retrieved.

math.DG↗
arXiv2017

Elastic flow interacting with a lateral diffusion process: The one-dimensional graph case

A finite element approach to the elastic flow of a curve coupled with a diffusion equation on the curve is analysed. Considering the graph case, the problem is weakly formulated and approximated with continuous linear finite elements, which is enabled thanks to second-order operator splitting. The error analysis builds up on previous results for the elastic flow. To obtain an error estimate for the quantity on the curve a better control of the velocity is required. For this purpose, a penalty approach is employed and then combined with a generalised Gronwall lemma. Numerical simulations support the theoretical convergence results. Further numerical experiments indicate stability beyond the parameter regime with respect to the penalty term which is covered by the theory.

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arXiv2016

The Łojasiewicz-Simon gradient inequality for open elastic curves

In this paper we consider the elastic energy for open curves in Euclidean space subject to clamped boundary conditions and obtain the Łojasiewicz-Simon gradient inequality for this energy functional. Thanks to this inequality we can prove that a (suitably reparametrized) solution to the associated $L^2$-gradient flow converges for large time to an elastica, that is to a critical point of the functional.

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arXiv2015

Curve shortening flow coupled to lateral diffusion

We present and analyze a semi-discrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve. More precisely, curve shortening flow with a forcing term that depends on a field defined on the curve is coupled with a diffusion equation for that field. The scheme is based on ideas of \cite{D99} for the curve shortening flow and \cite{DE07} for the parabolic equation on the moving curve. Additional estimates are required in order to show convergence, most notably with respect to the length element: While in \cite{D99} an estimate of its error was sufficient we here also need to estimate the time derivative of the error which arises from the diffusion equation. Numerical simulation results support the theoretical findings.

math.NA↗
arXiv2015

A Nested Variational Time Discretization for Parametric Anisotropic Willmore Flow

A variational time discretization of anisotropic Willmore flow combined with a spatial discretization via piecewise affine finite elements is presented. Here, both the energy and the metric underlying the gradient flow are anisotropic, which in particular ensures that Wulff shapes are invariant up to scaling under the gradient flow. In each time step of the gradient flow a nested optimization problem has to be solved. Thereby, an outer variational problem reflects the time discretization of the actual Willmore flow and involves an approximate anisotropic $L^2$-distance between two consecutive time steps and a fully implicit approximation of the anisotropic Willmore energy. The anisotropic mean curvature needed to evaluate the energy integrand is replaced by the time discrete, approximate speed from an inner, fully implicit variational scheme for anisotropic mean curvature motion. To solve the nested optimization problem a Newton method for the associated Lagrangian is applied. Computational results for the evolution of curves underline the robustness of the new scheme, in particular with respect to large time steps.

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arXiv2013

Willmore-Helfrich L^{2}-flow of curves with natural boundary conditions

We consider regular open curves in R^n with fixed boundary points and moving according to the L^{2}-gradient flow for a generalisation of the Helfrich functional. Natural boundary conditions are imposed along the evolution. More precisely, at the boundary the curvature vector is equal to the normal projection of a fixed given vector. A long-time existence result together with subconvergence to critical points is proven.

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