  This paper deals with the nonlinear phase field system
  \begin{equation*}
  \begin{cases}
  \partial_t (\theta +\ell \varphi) - \Delta\theta = f
  & \mbox{in}\ \Omega\times(0, T),
  \\[1mm]
  \partial_t \varphi - \Delta\varphi + \xi + \pi(\varphi)
  = \ell \theta,\ \xi\in\beta(\varphi)
  & \mbox{in}\ \Omega\times(0, T)
  \end{cases}
  \end{equation*} in a general domain $\Omega\subseteq\mathbb{R}^N$. Here $N
\in \mathbb{N}$, $T>0$, $\ell>0$, $f$ is a source term, $\beta$ is a maximal
monotone graph and $\pi$ is a Lipschitz continuous function. We note that in
the above system the nonlinearity $\beta+\pi$ replaces the derivative of a
potential of double well type. Thus it turns out that the system is a
generalization of the Caginalp phase field model and it has been studied by
many authors in the case that $\Omega$ is a bounded domain. However, for
unbounded domains the analysis of the system seems to be at an early stage. In
this paper we study the existence of solutions by employing a time
discretization scheme and passing to the limit as the time step $h$ goes to
$0$. In the limit procedure we face with the difficulty that the embedding
$H^1(\Omega) \hookrightarrow L^2(\Omega)$ is not compact in the case of
unbounded domains. Moreover, we can prove an interesting error estimate of
order $h^{1/2}$ for the difference between continuous and discrete solutions.
