Maxmimum norm estimates for ESFEM and time discretization
Quasilinear parabolic PDE on moving sufaces
Evolving surface finite element method (ESFEM)
Arbitrary Lagrange Eulerian maps (ALE) for ESFEM
Numerical Analysis of the mean curvature flow
Numerical experiments
The video above is a finite element solution for \((\vec X, u)\),
which solves a surface partial differential equation of the following type
\[
\partial^{\vec X(t)}_t u + div_{\vec X(t)}(\vec v) u - \Delta_{\vec X(t)} u = f,
\]
\[
\vec v - \alpha \Delta_{\vec X(t)} \vec v = g(u) \vec n_{\vec X(t)},
\]
where \(\vec X \colon M\times [0,T] \to \mathbb{R}^3\) describes the evolution
of the inital surface \(M\), with velocity
\(\vec v(\vec X,t)=\partial_t \vec X\), unit normal
\(n_{\vec X(t)}\), manifold time derivative
\(\partial^{\vec X(t)}_t u = \partial_t u(\vec X(t),t)\), manifold divergence
\(div_{\vec X(t)}\), Laplace-Beltrami operator
\(\Delta_{\vec X(t)}\), and where
\(u(t)\colon X(M,t)\to \mathbb{R}\) describes a chemical
concentration, which may promote or inhibit the growth of the surface.
Hence, the resulting surface deformates in a complicated fashion.
Snapshots from the following two videos
have been used in
[1].
The left-hand side works with the regularized velocity law above.
The right-hand side is regularized with a mean curvature term.
Source code is available on
Github. Documentation
is available here.