With this award the principal investigator will continue his studies into the numerical analysis of phase transitions and free boundary problems that arise in fluid dynamics and elasticity. In particular, he will analyze, design and implement adaptive finite element methods for the solution of the Phase Field Model and the Cahn-Hilliard equation. Special emphasis will be placed upon stability and error analyses in nonenergy norms, such as the maximum norm, and upon the efficiency and reliability of the resulting algorithms. Many phenomena in nature involve what are called free boundary problems, wherein a large part of the solution process is devoted to determining the location of a moving boundary between two or more different media. A good example is a weather forecast or the weather map in your newspaper. Here meteorologists try to predict the speed and location of fronts; in this context fronts are the words used for moving boundaries between warm and cold air, for example. With this the principal investigator will design numerical methods for finding the solution of some free boundary problems that come from the theory of elasticity. **
