Time discretization of a nonlinear phase field system in general domains

This paper deals with the nonlinear phase field system {partial derivative t(theta vertical bar l phi) - Delta theta = f in Omega x (0,T), partial derivative t phi - Delta phi + xi + pi(phi) = l theta, xi is an element of beta(phi) in Omega x (0,T) in a general domain Omega subset of R-d. Here d is an element of N, T > 0, l > 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) (sic) 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.