Mathematical Sciences: Numerical Approximation of Strongly Nonlinear Problems

获得该奖项后，首席研究员将继续研究流体动力学和弹性力学中出现的相变和自由边界问题的数值分析。特别是，他将分析、设计和实现自适应有限元方法来解决相场模型和Cahn-Hilliard方程。特别强调将放在稳定性和误差分析的非能量规范，如最大规范，并在效率和可靠性的结果算法。自然界中的许多现象涉及所谓的自由边界问题，其中求解过程的很大一部分致力于确定两种或多种不同介质之间移动边界的位置。天气预报或报纸上的天气图就是一个很好的例子。气象学家试图预测锋面的速度和位置；在这种情况下，锋面是用来表示冷热空气之间移动边界的词。在此基础上，主要研究者将设计一些数值方法来求解一些来自弹性理论的自由边界问题。**

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. **