Resumen
A nonisothermal phase field model for alloys with multiple phases and components is derived. The model allows for arbitrary phase diagrams. We relate the model to classical sharp interface models by formally matched asymptotic expansions. In addition we discuss several examples and relate our model to the ones already existing. Key words. phase field models, sharp interface models, phase transitions, partial differential equations, alloy systems, matched asymptotic expansions

Resumen
We consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations: \begin{alignat}{2} \textstyle{\frac{\partial u}{\partial t}} + \textstyle \frac{1}{2} \,\nabla . (u^2 \,\nabla [\sigma(v)]) - \textstyle \frac{1}{3}\, \nabla .(u^3 \,\nabla w) &= 0, &&%\quad\mbox{in} \;\;\Omega_T, \qquad %\hspace{2cm} \nonumber \\ w = - c \, \Delta u + a \, u^{-3} - \delta \, u^{-\nu}, \nonumber \\ \textstyle{\frac{\partial v}{\partial t}} + \nabla . (u\,v\,\nabla [\sigma(v)]) - \rho \,\Delta v - \textstyle \frac{1}{2}\, \nabla .(u^2\,v \,\nabla w) &= 0. &&%\quad\mbox{in} \;\;\Omega_T. \nonumber %\\ \end{alignat} The above models a surfactant-driven thin film flow in the presence of both attractive, a >0, and repulsive, $\delta >0$ with $\nu >3$, van der Waals forces, where u is the height of the film, v is the concentration of the insoluble surfactant monolayer, and $\sigma(v):=1-v$ is the typical surface tension. Here $\rho \geq 0$ and c>0 are the inverses of the surface Peclet number and the modified capillary number. In addition to showing stability bounds for our approximation, we prove convergence in one space dimension when $\rho >0$ and either $a=\delta=0$ or $\delta > 0$. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.