Space and Time Adaptivity for Moving and Free Boundary Problems

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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.