Nonlinear Analysis 63 (2005) e1143 – e1153 www.elsevier.com/locate/na Modelling phase transitions in alloys Maria Gokieli a , ∗ , Leszek Marcinkowski b, 1 a ICM, Warsaw University, Zwirki i Wigury 93, 02-089 Warsaw, Poland b Department of Mathematics,Warsaw University, Banacha 2, 02-097Warsaw, Poland Abstract We present a numerical method for solving a system of two nonlinear evolution equations of order four and two, known as the Cahn-Hilliard/Allen-Cahn system. The simulation results give a first intuition about the stationary states form and stability.  2005 Elsevier Ltd. All rights reserved. Keywords: Evolution equations; Finite element; Attractor 1. Introduction Two main kinds of phase transition phenomena in metallic alloys are widely known (cf. e.g. [8]). The first one is phase separation, including processes known as spinodal decomposition which is very quick, and grain growth or Ostwald ripening, which is slow. This means appearance, then growth of grained structure in the metal, under quenching. By grains we understand domains occupied by only one component of the alloy. The second is ordering, i.e. appearance of a crystallic, ordered lattice structure, occupied in a regular way by both components, at the atomic level. Phase separation has been first modelled by Van der Waals [15]; in 1957, Cahn and Hilliard [4] have re-discovered his model and since, it has been intensively studied. See [13,7,2] for mathematical analysis and [3,6] for numerics. Also, the results obtained in [1] in one dimension suggest that the whole process is an alternance of several very rapid and very slow stages. ∗ Corresponding author. Fax: +48 22 874 9115. 1 Partially supported by the Polish State research Grant number 2 PO3A 005 24. 0362-546X/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.03.090