  We introduce unconditionally stable finite element approximations for a phase
field model for solidification, which take highly anisotropic surface energy
and kinetic effects into account. We hence approximate Stefan problems with
anisotropic Gibbs--Thomson law with kinetic undercooling, and quasi-static
variants thereof. The phase field model is given by {align*} \vartheta\,w_t +
\lambda\,\varrho(\varphi)\,\varphi_t & = \nabla \,.\, (b(\varphi)\,\nabla\, w)
\,, \cPsi\,\tfrac{a}\alpha\,\varrho(\varphi)\,w & =
\epsilon\,\tfrac\rho\alpha\,\mu(\nabla\,\varphi)\,\varphi_t -\epsilon\,\nabla
\,.\, A'(\nabla\, \varphi) + \epsilon^{-1}\,\Psi'(\varphi) {align*} subject to
initial and boundary conditions for the phase variable $\varphi$ and the
temperature approximation $w$. Here $\epsilon > 0$ is the interfacial
parameter, $\Psi$ is a double well potential, $\cPsi = \int_{-1}^1
\sqrt{2\,\Psi(s)}\;{\rm d}s$, $\varrho$ is a shape function and
$A(\nabla\,\varphi) = \tfrac12\,|\gamma(\nabla\,\varphi)|^2$, where $\gamma$ is
the anisotropic density function. Moreover, $\vartheta \geq 0$, $\lambda > 0$,
$a > 0$, $\alpha > 0$ and $\rho \geq 0$ are physical parameters from the Stefan
problem, while $b$ and $\mu$ are coefficient functions which also relate to the
sharp interface problem.
  On introducing the novel fully practical finite element approximations for
the anisotropic phase field model, we prove their stability and demonstrate
their applicability with some numerical results.
