10:40
25 mins
|
On modified diffuse-interface approximations of Willmore-flow for colliding interfaces
Andreas Raetz, Matthias Roeger
Abstract: In this talk on a joint work with M. Röger (TU Dortmund), we are concerned with diffuse-interface approximations
for Willmore flow. As observed in previous numerical results of standard diffuse-interface models for colliding one dimensional interfaces, in such a scenario evolutions towards interfaces with corners can occur contradicting the corresponding sharp-interface model.
We propose a new modified diffuse-interface energy which is incorporated into the evolution law and avoids the formation of interfaces with corners. We present formal asymptotics and numerical simulations to justify the modification. Moreover, various application examples are presented.
|
11:05
25 mins
|
Uniqueness for a second order gradient flow of elastic networks
Paola Pozzi, Matteo Novaga
Abstract: In a previous work by the authors a second order gradient flow of the p-elastic energy for a planar theta-network of three curves
with fixed lengths was considered and a weak solution of the flow was constructed by means of an implicit variational scheme.
Long-time existence of the evolution and convergence to a critical point of the energy were shown. The purpose of this note is to prove uniqueness of the weak solution when p=2.
|
11:30
25 mins
|
Finite element approximation of a system coupling curve evolution with prescribed normal contact to a fixed boundary to reaction-diffusion on the curve
James Van Yperen, Vanessa Styles
Abstract: We consider a finite element approximation for a system consisting of the evolution of a curve evolving by forced curve shortening flow coupled to a reaction-diffusion equation on the evolving curve. The curve evolves inside a given domain and meets the boundary of that domain orthogonally.
The scheme for the coupled system is based on the schemes derived in Barrett, Deckelnick and Styles (2017) and Deckelnick and Elliott (1998). We present numerical experiments and show the experimental order of convergence of the approximation.
|
11:55
25 mins
|
Finite element analysis for a diffusion equation on an evolving domain driven by a harmonic velocity law
Dominik Edelmann
Abstract:
We present a convergence analysis for an evolving finite element semi-discretization of a parabolic partial differential equation on an evolving bulk domain. The boundary of the domain evolves with a given velocity, which is then harmonically extended to the bulk by solving a Poisson equation. The numerical solution to the diffusion equation depends on the numerical solution to this bulk evolution itself, which yields the time-dependent computational domain for the finite element method. The stability analysis works with the matrix-vector formulation of the semi-discretization only and does not require geometric arguments, which enter the proof in consistency estimates. Combining the obtained stability and consistency estimates yields convergence in H1 of optimal order. We will also present various numerical experiments.
|
