21 research outputs found
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 generalized Gronwall lemma. Numerical simulations support the theoretical convergence results. Further numerical experiments indicate stability beyond the parameter regime with respect to the penalty term that is covered by the theory
Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds
We present variational approximations of boundary value problems for
curvature flow (curve shortening flow) and elastic flow (curve straightening
flow) in two-dimensional Riemannian manifolds that are conformally flat. For
the evolving open curves we propose natural boundary conditions that respect
the appropriate gradient flow structure. Based on suitable weak formulations we
introduce finite element approximations using piecewise linear elements. For
some of the schemes a stability result can be shown. The derived schemes can be
employed in very different contexts. For example, we apply the schemes to the
Angenent metric in order to numerically compute rotationally symmetric
self-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes
to compute geodesics that are relevant for optimal interface profiles in
multi-component phase field models.Comment: 42 pages, 21 figure