91 citations · 747 across the 46 of their papers we have counts for
47 papers
An energy-stable parametric finite element framework for surface diffusion with moving contact lines on curved substrates
Harald Garcke, Robert Nürnberg, Rong Tang
We develop energy-stable parametric finite element methods for surface diffusion of solid films with moving contact lines on curved substrates. Discretizing both the film surface a…
A filtered time stepping scheme for curve shortening flow for open and closed curves
Klaus Deckelnick, Robert Nürnberg
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 c…
A unified energy-stable finite element approximation for evolving fluidic biomembranes
Harald Garcke, Robert Nürnberg, Quan Zhao
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…
A Parametric Finite Element Approach for an Anisotropic Multi-Phase Mullins-Sekerka Problem with Kinetic Undercooling
Tokuhiro Eto, Harald Garcke, Robert Nürnberg
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 evolv…
On the basin of attraction for the free boundary free elastic flow
Klaus Deckelnick, Hans-Christoph Grunau, Robert Nürnberg +2
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…
A parametric finite element method for the incompressible Navier--Stokes equations on an evolving surface
Harald Garcke, Robert Nürnberg
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 s…
An energy-stable parametric finite element method for Willmore flow with normal-tangential velocity splitting
Harald Garcke, Robert Nürnberg, Quan Zhao
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 s…
A variational front-tracking method for multiphase flow with triple junctions
Harald Garcke, Robert Nürnberg, Quan Zhao
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 ne…
Parametric finite element approximation of two-phase Navier--Stokes flow with viscoelasticity
Harald Garcke, Robert Nürnberg, Dennis Trautwein
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 Navi…
A finite element method for anisotropic crystal growth on surfaces
Harald Garcke, Robert Nürnberg
Phase transition problems on curved surfaces can lead to a panopticon of fascinating patterns. In this paper we consider finite element approximations of phase field models with a…
Finite element schemes with tangential motion for fourth order geometric curve evolutions in arbitrary codimension
Klaus Deckelnick, Robert Nürnberg
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 spe…
Discrete anisotropic curve shortening flow in higher codimension
Klaus Deckelnick, Robert Nürnberg
We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in , . The reformulation hinges on a suitable ma…
A structure-preserving finite element method for the multi-phase Mullins-Sekerka problem with triple junctions
Tokuhiro Eto, Harald Garcke, Robert Nürnberg
We consider a sharp interface formulation for the multi-phase Mullins-Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of…
Arbitrary Lagrangian-Eulerian finite element approximations for axisymmetric two-phase flow
Harald Garcke, Robert Nürnberg, Quan Zhao
We analyze numerical approximations for axisymmetric two-phase flow in the arbitrary Lagrangian-Eulerian (ALE) framework. We consider a parametric formulation for the evolving flui…
Unfitted finite element methods for axisymmetric two-phase flow
Harald Garcke, Robert Nürnberg, Quan Zhao
We propose and analyze unfitted finite element approximations for the two-phase incompressible Navier--Stokes flow in an axisymmetric setting. The discretized schemes are based on…
Structure-preserving discretizations of two-phase Navier-Stokes flow using fitted and unfitted approaches
Harald Garcke, Robert Nürnberg, Quan Zhao
We consider the numerical approximation of a sharp-interface model for two-phase flow, which is given by the incompressible Navier-Stokes equations in the bulk domain together with…
A diffuse-interface approach for solid-state dewetting with anisotropic surface energies
Harald Garcke, Patrik Knopf, Robert Nürnberg +1
We present a diffuse-interface model for the solid-state dewetting problem with anisotropic surface energies in for . The introduced model consists of…
Deep learning for gradient flows using the Brezis-Ekeland principle
Laura Carini, Max Jensen, Robert Nürnberg
We propose a deep learning method for the numerical solution of partial differential equations that arise as gradient flows. The method relies on the Brezis--Ekeland principle, whi…
An unconditionally stable finite element scheme for anisotropic curve shortening flow
Klaus Deckelnick, Robert Nürnberg
Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method u…
Discrete hyperbolic curvature flow in the plane
Klaus Deckelnick, Robert Nürnberg
Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by Gurtin a…
A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters
Weizhu Bao, Harald Garcke, Robert Nürnberg +1
We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, wi…
Overhang penalization in additive manufacturing via phase field structural topology optimization with anisotropic energies
Harald Garcke, Kei Fong Lam, Robert Nürnberg +1
A phase field approach for structural topology optimization with application to additive manufacturing is analyzed. The main novelty is the penalization of overhangs (regions of th…
A structure preserving front tracking finite element method for the Mullins--Sekerka problem
Robert Nürnberg
We introduce and analyse a fully discrete approximation for a mathematical model for the solidification and liquidation of materials of negligible specific heat. The model is a two…
Volume-preserving parametric finite element methods for axisymmetric geometric evolution equations
Weizhu Bao, Harald Garcke, Robert Nurnberg +1
We propose and analyze volume-preserving parametric finite element methods for surface diffusion, conserved mean curvature flow and an intermediate evolution law in an axisymmetric…