65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems
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- 2011 (1)
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- multigrid (1)
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- C (1)
We consider anisotropic Allen--Cahn equations with interfacial energy
induced by an anisotropic surface energy density $\gamma$.
Assuming that $\gamma$
is positive, positively homogeneous of degree one,
strictly convex in tangential directions to the unit sphere,
and sufficiently smooth, we show stability of
various time discretizations. In particular,
we consider a fully implicit and a linearized time discretization
of the interfacial energy combined with implicit
and semi-implicit time discretizations
of the double-well potential. In the semi-implicit variant,
concave terms are taken explicitly.
The arising discrete spatial problems are solved by
globally convergent truncated nonsmooth Newton multigrid methods.
Numerical experiments show the accuracy of the different
discretizations.
We also illustrate that pinch-off under anisotropic
mean curvature flow is no longer frame invariant,
but depends on the orientation of the initial configuration.