arXiv subjects

Robert Nürnberg

Publications and source records attributed to Robert Nürnberg.

At least 19 recordsLinked to original sources

arXiv2026

A filtered time stepping scheme for curve shortening flow for open and closed curves

We propose a filtered time stepping finite element scheme for curve shortening flow of open and closed curves in arbitrary codimension that is second-order accurate in time. Open curves are assumed to evolve inside a given domain $Ω\subset \mathbb R^n$, $n\geq2$, and meet the external boundary $\partialΩ$ orthogonally. We prove optimal error bounds for the $L^2$-- and $H^1$--norms. In practice only a single linear system needs to be solved at each time step. Numerical experiments confirm the accuracy and practicality of the introduced method, including an asymptotic equidistribution property.

math.NA↗

arXiv2026

On the basin of attraction for the free boundary free elastic flow

The free boundary free elastic flow is the steepest descent gradient flow for the elastic energy of curves meeting parallel lines perpendicularly. In this article we prove that the straight line has, measured in Euler's scale-invariant bending energy, a basin of attraction at least to the level $1.9615\, π$. We show that our method of proof cannot be pushed to the previously conjectured level $2π$, and in addition present numerical evidence that this conjecture may in fact be false.

math.AP↗

arXiv2026

A parametric finite element method for the incompressible Navier--Stokes equations on an evolving surface

In this paper we consider the numerical approximation of the incompressible surface Navier--Stokes equations on an evolving surface. For the discrete representation of the moving surface we use parametric finite elements of degree $\ell \geq 2$. In the semidiscrete continuous-in-time setting we are able to prove a stability estimate that mimics a corresponding result for the continuous problem. Some numerical results, including a convergence experiment, demonstrate the practicality and accuracy of the proposed method.

math.NA↗

arXiv2026

A unified energy-stable finite element approximation for evolving fluidic biomembranes

We present a unified finite element method for the dynamics of fluidic biomembranes. The model is governed by the Navier--Stokes equations in the bulk coupled to the surface Navier--Stokes equations on the evolving biomembrane surface, with bending forces arising from the Willmore energy. By allowing the bulk mesh velocity to be independent of the fluid velocity and permitting a free tangential surface velocity, we are able to derive a unified weak formulation of the coupled bulk-surface Navier--Stokes system. To address the bending force, we consider an evolution equation for the curvature and propose a surface arbitrary Lagrangian--Eulerian (ALE) weak formulation. Discretization with either fitted or unfitted finite elements leads to well-posed fully discrete linear schemes that are unconditionally energy stable. We present a variety of numerical examples to demonstrate the favourable properties of the proposed methods.

math.NA↗

arXiv2026

A Parametric Finite Element Approach for an Anisotropic Multi-Phase Mullins-Sekerka Problem with Kinetic Undercooling

We consider a sharp interface formulation for an anisotropic multi-phase Mullins-Sekerka problem with kinetic undercooling. The flow is characterized by a cluster of surfaces evolving such that the total surface energy plus a weighted sum of the volumes of the enclosed phases decreases in time. Upon deriving a suitable variational formulation, we introduce a fully discrete unfitted finite element method. In this approach, the approximations of the moving interfaces are independent of the triangulations used for the equations in the bulk. Our method can be shown to be unconditionally stable. Several numerical examples demonstrate the capabilities of the introduced method. In particular, it is demonstrated that the evolution of multiple ice crystals with junctions can be modeled using the proposed approach.

math.NA↗

arXiv2026

On a Mullins-Sekerka model for the growth of active droplets modelling protocells: Stability analysis and numerical computations

Mullins-Sekerka models with chemical reactions can lead to scenarios where droplets grow, become unstable, split, grow and undergo further division. These grow and division cycles have been proposed as a model for protocells and are believed to play a fundamental role in living systems by providing chemical compartments which are important in the organization of living systems. This paper analyses chemically active Mullins-Sekerka models. Existence of radially symmetric solutions is shown and a detailed stability analysis in radial as well as planar situations is given. In particular, we also analyze multilayered solutions leading to shell-type situations. Finally, we introduce a numerical method based on a parametric finite element approach that explicitly accounts for topological changes, thereby allowing for droplet splitting and merging. Several numerical simulations verify the findings of the theoretical stability analysis and show complex dynamical behavior, including multiple instabilities, splittings of droplets and appearance of shell-type solutions.

math.AP↗

arXiv2025

An energy-stable parametric finite element method for Willmore flow with normal-tangential velocity splitting

We propose and analyze an energy-stable fully discrete parametric approximation for Willmore flow of hypersurfaces in two and three space dimensions. We allow for the presence of spontaneous curvature effects and for open surfaces with boundary. The presented scheme is based on a new geometric partial differential equation (PDE) that combines an evolution equation for the mean curvature with a separate equation that prescribes the tangential velocity. The mean curvature is used to determine the normal velocity within the gradient flow structure, thus guaranteeing an unconditional energy stability for the discrete solution upon suitable discretization. We introduce a novel weak formulation for this geometric PDE, in which different types of boundary conditions can be naturally enforced. We further discretize the weak formulation to obtain a fully discrete parametric finite element method, for which well-posedness can be rigorously shown. Moreover, the constructed scheme admits an unconditional stability estimate in terms of the discrete energy. Extensive numerical experiments are reported to showcase the accuracy and robustness of the proposed method for computing Willmore flow of both curves in $\mathbb{R}^2$ and surfaces in $\mathbb{R}^3$.

math.NA↗

arXiv2025

A parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions

In this study, we propose a parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions. This model describes the energy-driven motion of a surface cluster whose distributional solution was studied by Garcke and Sturzenhecker. We approximate the weak formulation of this sharp interface model by an unfitted finite element method that uses parametric elements for the representation of the moving interfaces. We establish existence and uniqueness of the discrete solution and prove unconditional stability of the proposed scheme. Moreover, a modification of the original scheme leads to a structure-preserving variant, in that it conserves the discrete analogue of a quantity that is preserved by the classical solution. Some numerical results demonstrate the applicability of our introduced schemes.

math.NA↗

arXiv2025

Stable fully discrete finite element methods with BGN tangential motion for Willmore flow of planar curves

We propose and analyze stable finite element approximations for Willmore flow of planar curves. The presented schemes are based on a novel weak formulation which combines an evolution equation for curvature with the curvature formulation originally proposed by Barrett, Garcke and Nürnberg (BGN) in \cite{BGN07}. Under discretization in space with piecewise linear elements this leads to a stable continuous-in-time semidiscrete scheme, which retains the equidistribution property from the BGN methods. Furthermore, two fully discrete schemes can be shown to satisfy unconditional energy stability estimates. Numerical examples are presented to showcase the good properties of the introduced schemes, including an asymptotic equidistribution of vertices.

math.NA↗

arXiv2025

Second order in time finite element schemes for curve shortening flow and curve diffusion

We prove optimal error bounds for a second order in time finite element approximation of curve shortening flow in possibly higher codimension. In addition, we introduce a second order in time method for curve diffusion. Both schemes are based on variational formulations of strictly parabolic systems of partial differential equations that feature a tangential velocity which under discretization is beneficial for the mesh quality. In each time step only two linear systems need to be solved. Numerical experiments demonstrate second order convergence as well as asymptotic equidistribution.

math.NA↗

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.

math.AP↗

arXiv2025

Isoparametric finite element methods for mean curvature flow and surface diffusion

We propose higher-order isoparametric finite element approximations for mean curvature flow and surface diffusion. The methods are natural extensions of the piecewise linear finite element methods introduced by Barrett, Garcke, and Nürnberg (BGN) in a series of papers in 2007 and 2008. The proposed schemes exhibit unconditional energy stability and inherit the favorable mesh quality of the original BGN methods. Moreover, in the case of surface diffusion we present structure-preserving higher-order isoparametric finite element methods. In addition to being unconditionally stable, these also conserve the enclosed volume. Extensive numerical results demonstrate the higher-order spatial accuracy, the unconditional energy stability, the volume preservation for surface diffusion, and the good mesh quality.

math.NA↗

arXiv2025

Stable fully practical finite element methods for axisymmetric Willmore flow

We consider fully discrete numerical approximations for axisymmetric Willmore flow that are unconditionally stable and work reliably without remeshing. We restrict our attention to surfaces without boundary, but allow for spontaneous curvature effects. The axisymmetric setting allows us to formulate our schemes in terms of the generating curve of the considered surface. We propose a novel weak formulation, that combines an evolution equation for the surface's mean curvature and the curvature identity of the generating curve. The mean curvature is used to describe the gradient flow structure, which enables an unconditional stability result for the discrete solutions. The generating curve's curvature, on the other hand, describes the surface's in-plane principal curvature and plays the role of a Lagrange multiplier for an equidistribution property on the discrete level. We introduce two fully discrete schemes and prove their unconditional stability. Numerical results are provided to confirm the convergence, stability and equidistribution properties of the introduced schemes.

math.NA↗

arXiv2025

Finite element schemes with tangential motion for fourth order geometric curve evolutions in arbitrary codimension

We introduce novel finite element schemes for curve diffusion and elastic flow in arbitrary codimension. The schemes are based on a variational form of a system that includes a specifically chosen tangential motion. We derive optimal $L^2$- and $H^1$-error bounds for continuous-in-time semidiscrete finite element approximations that use piecewise linear elements. In addition, we consider fully discrete schemes and, in the case of curve diffusion, prove unconditional stability for it. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bounds. The presented simulations suggest that the tangential motion leads to equidistribution in practice.

math.NA↗

arXiv2025

On a Cahn-Hilliard equation for the growth and division of chemically active droplets modeling protocells

The Cahn-Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn-Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

math.AP↗

arXiv2025

On the numerical approximation of hyperbolic mean curvature flows for surfaces

The paper addresses the numerical approximation of two variants of hyperbolic mean curvature flow of surfaces in $\mathbb R^3$. For each evolution law we propose both a finite element method, as well as a finite difference scheme in the case of axially symmetric surfaces. We present a number of numerical simulations, including convergence tests as well as simulations suggesting the onset of singularities.

math.NA↗

arXiv2024

A variational front-tracking method for multiphase flow with triple junctions

We present and analyze a variational front-tracking method for a sharp-interface model of multiphase flow. The fluid interfaces between different phases are represented by curve networks in two space dimensions (2d) or surface clusters in three space dimensions (3d) with triple junctions where three interfaces meet, and boundary points/lines where an interface meets a fixed planar boundary. The model is described by the incompressible Navier--Stokes equations in the bulk domains, with classical interface conditions on the fluid interfaces, and appropriate boundary conditions at the triple junctions and boundary points/lines. We propose a weak formulation for the model, which combines a parametric formulation for the evolving interfaces and an Eulerian formulation for the bulk equations. We employ an unfitted discretization of the coupled formulation to obtain a fully discrete finite element method, where the existence and uniqueness of solutions can be shown under weak assumptions. The constructed method admits an unconditional stability result in terms of the discrete energy. Furthermore, we adapt the introduced method so that an exact volume preservation for each phase can be achieved for the discrete solutions. Numerical examples for three-phase flow and four-phase flow are presented to show the robustness and accuracy of the introduced methods.

math.NA↗

arXiv2024

Parametric finite element approximation of two-phase Navier--Stokes flow with viscoelasticity

In this work, we present a parametric finite element approximation of two-phase Navier-Stokes flow with viscoelasticity. The free boundary problem is given by the viscoelastic Navier-Stokes equations in the two fluid phases, connected by jump conditions across the interface. The elasticity in the fluids is characterised using the Oldroyd-B model with possible stress diffusion. The model was originally introduced to approximate fluid-structure interaction problems between an incompressible Newtonian fluid and a hyperelastic neo-Hookean solid, which are possible limit cases of the model. We approximate a variational formulation of the model with an unfitted finite element method that uses piecewise linear parametric finite elements. The two-phase Navier-Stokes-Oldroyd-B system in the bulk regions is discretised in a way that guarantees unconditional solvability and stability for the coupled bulk-interface system. Good volume conservation properties for the two phases are observed in the case where the pressure approximation space is enriched with the help of an XFEM function. We show the applicability of our method with some numerical results.

math.NA↗