A Finite Element Algorithm
for the
Mean Curvature Flow
of
Level Sets
Let the hypersurface
be given as the
zero level set of a smooth function 
.
Searching for the evolution of 
by mean curvature, we investigate
the following PDE:
The zero level set of the (weak) solution 
of (1)
moves according to its mean curvature (as well as all other level sets
of 
), at least in regions where is smooth and 
is
non-vanishing. The family 
of zero level sets
of 
is called `generalized evolution of by mean curvature'.
(See Evans and Spruck, [1],[2],[3]) We introduce an adaptive Finite Element algorithm to approximate this
generalized evolution of the initial hypersurface .
Restricting ourselves to a polygonal domain 
we use the
following 
-regularization of the nonlinear degenerated
PDE (1)
A corresponding Finite Element semi-discretization with grid size 
is given
by:
with a suitable Finite Element space 
. An additional time discretization
leads to a nonlinear system of equations which is linearized by application
of a modified Newton's method. This gives a nonsymmetric linear system.
with errors
and grid size 
.
The following figures show the EOCs for (a) the shrinking circle and (b) the
shrinking ball. Figure (c) shows the computed radius of the circle for different grid sizes
compared with the exact radius.
may develop an interior, even if the initial set had none.
One example may be the union of two different axes in 
. The following
figures (g),
(h), and
(i) show
the level sets of the computed function
at different timesteps.
Here may develop an interior: After some timesteps
is very flat around the zero level set.
The shape of this nearly constant region of looks like the shape
Evans and Spruck expected for 
.
.
A breaking dumbbell
(here is another image of this dumbbell),
two tori.
The thin torus shrinks to a circle, the fat one becomes a ball before it
vanishes.
J. Michael Fried, Universität Freiburg, Germany
