Time discretization of a nonlinear phase field system in general domains

Abstract

This paper deals with the nonlinear phase field system \begin{equation*} \begin{cases} \partial_t (θ+\ell φ) - Δθ= f & \mbox{in}\ Ω\times(0, T), \\[1mm] \partial_t φ- Δφ+ ξ+ π(φ) = \ell θ,\ ξ\inβ(φ) & \mbox{in}\ Ω\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.