  We are concerned with the mathematical modeling of the polarization process
in ferroelectric media. We assume that this dissipative process is governed by
two constitutive functions, which are the free energy function and the
dissipation function. The dissipation function, which is closely connected to
the dissipated energy, is usually non-differentiable. Thus, a minimization
condition for the overall energy includes the subdifferential of the
dissipation function. This condition can also be formulated by way of a
variational inequality in the unknown fields strain, dielectric displacement,
remanent polarization and remanent strain. We analyze the mathematical
well-posedness of this problem. We provide an existence and uniqueness result
for the time-discrete update equation. Under stronger assumptions, we can prove
existence of a solution to the time-dependent variational inequality. To solve
the discretized variational inequality, we use mixed finite elements, where
mechanical displacement and dielectric displacement are unknowns, as well as
polarization (and, if included in the model, remanent strain). It is then
possible to satisfy Gauss' law of zero free charges exactly. We propose to
regularize the dissipation function and solve for all unknowns at once in a
single Newton iteration. We present numerical examples gained in the open
source software package Netgen/NGSolve.
