Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds
| Autor: | Harald Garcke, Robert Nürnberg |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Mathematics - Differential Geometry
Mean curvature flow Geodesic Curve-shortening flow Applied Mathematics Mathematical analysis Numerical Analysis (math.NA) Weak formulation Curvature Computational Mathematics Differential Geometry (math.DG) Flow (mathematics) FOS: Mathematics Boundary value problem Mathematics::Differential Geometry Mathematics - Numerical Analysis Balanced flow Mathematics |
| Popis: | 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. 42 pages, 21 figures |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|