Ralf Kornhuber : Monotone multigrid for Allen-Cahn equations
Solidification of liquids or phase separation,
e.g.,in binary alloys is frequently described mathematically by phase field models.
The micromolecular phenomena at the interface
are represented by an order parameter.
We aim at the construction of robust
multigrid solvers for large algebraic systems
as resulting from implicit time discretization of
Allen-Cahn equations (non-conserved order parameter)
and Cahn-Hilliard equations (conserved order parameter).
Robustness means that convergence behavior should
be insensitive not only with respect to
discretization parameters such as mesh size or time step
but also with respect to relevant parameters
of the continuous problem such as the amount of interfacial energy or temperature.
In this lecture we concentrate on multigrid algorithms
for scalar and vector-valued Allen-Cahn equations
with obstacle potential. The construction is based on
successive minimization of energy with respect to a
sequence of d-dimensional subspaces representing a
scale of frequencies. Suitable selection of these
subspaces together with monotone reduction of energy
provides global convergence.
Robust extensions to logarithmic free energy is
obtained by so-called constrained Newton linearization.
Possible variants for Cahn-Hilliard equations
are also discussed.