Space and Time Adaptivity for Moving and Free Boundary Problems

该奖项是根据2009年美国复苏和再投资法案（公法111 - 5）资助的。充分的数值处理自由边界问题表现出不同的尺度是一个可怕的数学和计算的挑战，几何形状起着至关重要的作用。现代算法应该能够优化和平衡的计算工作，以捕捉小尺度，而不会过度解决其他生产特别有效的接口几何形状的描述。因此，在这种情况下，适应性程序是至关重要的，但却苦于缺乏数学理解。本研究提出设计，测试和分析适用于自由边界问题的时空自适应算法。在生物物理学（如生物膜和心血管系统）和材料科学（如晶体表面弛豫，注塑粘弹性流动，和微器件设计）的具体情况下解决。所有这些都有一个共同点，即界面之间的有趣耦合以及由偏微分方程控制的一些感兴趣的量。可变形域在许多研究领域中无处不在，但仍然是一个严重的计算挑战。该项目提出了强大而高效的算法，特别针对这些问题进行了调整，从而实现了现实的预测和更深入的理解。其结果将有利于许多不同的研究领域。事实上，所讨论的应用在诸如人类细胞形态学、通信技术、纳米技术和高性能计算等战略主题中是有意义的。该项目是合作和跨学科的，涉及美国和国外的一些科学家。在教育方面投入了大量的努力。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The adequate numerical treatment of free boundary problems exhibiting disparate scales is a formidable mathematical and computational challenge where the geometry plays a crucial role. Modern algorithms should be able to optimize and balance the computational effort to capture small scales without over-resolving the others producing in particular efficient interface geometry description. Adaptive procedures in this context are thus critical but yet suffering from their lake of mathematical understanding. The present research proposes to design, test and analyze space-time adaptive algorithms suited for free boundary problems. Particular instances in biophysics (such as biomembranes and cardiovascular system) and material sciences (such as crystal surfaces relaxation, injection molding viscoelastic flow, and micro-devices design) are addressed. All have in common the intriguing coupling between interfaces and some quantities of interest governed by partial differential equations.Deformable domains are ubiquitous in several areas of research but are still a serious computational challenge. This project proposes robust and efficient algorithms particularly tuned for such problems leading to realistic predictions and deeper understanding. The outcome will benefit many differentresearch areas. In fact, the applications discussed are of interest in strategic topics such as human cells morphology, communication technology, nanotechnology, and high performance computing. This project is collaborative and interdisciplinary involving a number of scientists in the US and abroad. A substantial effort is devoted to education.