Parametric finite element approximations of curvature driven interface evolutions
arXiv:1903.09462 · doi:10.1016/bs.hna.2019.05.002
Abstract
Parametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature. The approaches discussed, in contrast to many other methods, have good mesh properties that avoid mesh coalescence and very non-uniform meshes. Mean curvature flow, surface diffusion, anisotropic geometric flows, solidification, two-phase flow, Willmore and Helfrich flow as well as biomembranes are treated. We show stability results as well as results explaining the good mesh properties.
144 pages, 7 figures. This review article will appear in HNA vol 21
References in corpus (4)
- Numerical computations of facetted pattern formation in snow crystal growth
- Finite element methods for fourth order axisymmetric geometric evolution equations
- Variational discretization of axisymmetric curvature flows
- The Phase Field Method for Geometric Moving Interfaces and Their Numerical Approximations
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